WATHEMATICS LIBRARY 
THE UN IVERSITY 


OF ILLINOIS 
LIBRARY 


The 
Frank Hall Collection 
of arithmetics, 
presented by Professor 
H, L. Rietz of the 
University of Iowa, 


513 
Gre 
IGECL 


h 


eRe MATHEMATICAL SERIES. 


AN 


ELEMENTARY ARITHMETIC. 


BY 


G. P“QUACKENBOS, A. M.,, 


“AN ENGLISH GRAMMAR;”’ “‘FIRST LESSONS IN COMPOSITION; “ ADVANCED 
COURSE OF COMPOSITION AND RHETORIC;” ‘A NATURAL PHILOSO- 
PHY;” “ILLUSTRATED SCHOOL HISTORY OF THE UNITED 
STATES;” “PRIMARY UISTORY OF THE UNITED 
BTATES;” ETO. 


UPON THE BASIS OF THE WORKS OF 


GEO. R. PERKINS, LL.D. 


NEW YORK: : 
D. APPLETON AND COMPANY, 
4438 & 445 BROADWAY. 
1866. 


SHl2Z) Mu 


22 
\ SGC WATHEMATICS LipRagy 


2 he PA: CR 


Tas volume is intended to follow our Primary Arithmetic, or 
that of any other series, or may be used as a first book with beginners 
that are not too young. It goes over the ground covered by the 
Primary, but in a style suited to minds somewhat more mature, en- 
larging on the subjects there treated, and introducing the pupil to 
many new ones. SBesides the four fundamental operations, it gives 
a comprehensive view of Fractions, Federal Money, Reduction, and 
the Compound Rules, presenting under each a large collection of 
sums, in every variety, not too difficult, but so constructed as to 
require the pupil to think, and thus make the performance intelligent 
and not mechanical. 

Convinced that too much theory and rule embarrass the young 
pupil, the author has in this respect sought to strike a happy mean,— 
presenting necessary explanations, but in few words; giving example 
sometimes the precedence over precept, and making rules intelligible 


by means of preliminary illustrations. Definitions are made brief 


and simple. Technical terms unnecessary at this stage of progress 
are avoided. The difficulties of beginners being appreciated, it is 
believed that they are here so met as to save the teacher the annoy- 
ance of constant demands for explanation. 

In arrangement we trust some gain will be apparent; particularly 
in Compound Numbers, where, in stead of presenting the Tables in 
a body, to be confounded together in the pupil’s mind, we imme- 
diately apply each Table, as soon as learned, in appropriate exercises, 
either mental or written. Attention is also invited to the inductive 
method used in developing the several subjects. 

The teacher is requested to see that every principle is mastered 
as the pupil advances. A single defective link makes a whole chain 
worthless. If this suggestion is attended to, it is believed that the 
present work will make the young student thoroughly acquainted 
with the subjects it embraces, and properly prepare him for the-next 
number of the series, THE PRacTICAL ARITHMETIC, 


New Yors, August 6, 1863, 


405945 


CONTENTS. 


PAGE 

Wouart ARITHMETIC IS, . ‘ : : : 5 , 5 
Noration, ; ‘ ‘ . ; : : : ; 6 
NUMERATION, : d ; : ° ; : ; “e616 
ADDITION, REE eam iy 
SUBTRACTION, : 5 : . ° : . * Cases 
MULTIPLICATION, : : ‘ ; , ‘ . . 37 
DIVISION, . A , F ° ; = , so 260 
Short Division, . e : i : : é E 53 
Long Division, , “lie ME ee TG 
FRACTIONS, : ; : 4 s : : gh 66 
Reduction of Fractions, . : ‘ : : Oar 
Addition of Fractions, ‘ ; ‘ : staat 78 
Subtraction of Fractions, : : of ues ; MBL 
Multiplication of Fractions, : A ser ; : 83 
Division of Fractions, . é : : : ‘ seer 
Freperat Money, : : A ‘ : , : 95 
Addition of Federal Money, : ‘ : > <n eae hee 
Subtraction of Federal Money, . ; : 3 - 99 


Multiplication of Federal Money, . : R ‘ EAN AQT 
Division of Federal Money, ; : : : : 102 
REDUCTION, . : " , ° . . : - 104 
Reduction Descending, S z . : < . 104 
Reduction Ascending, . : ‘ z Pay aid 06 


Compound NUMBERS, é < : ; : . : 108 
CompounD ADDITION, . ‘ ; : ; ° . ee eA!) 
Compounp SUBTRACTION, . : . ‘ : : A 183 
Compounp MULTIPLICATION, . ; , 2 : F agg 
Compounp Division, . : 4 : : : ° : 139 


MISCELLANEOUS SUMS, . é ; : : : . |» Fee 


ELEMENTARY ARITHMETIC. 


Wuar Arirumetic Is. 


1. Wr commence with onz. We have one head, 
one mouth, one body. 

One, a single thing, is called a Unit. 

2, A unit joined to another unit, makes two. 
We have two eyes, two hands, two feet. 

Another unit joined to two, makes turer. Each 
of our fingers has three joints. 

Another unit joined to three, makes rour. 

So we may go on.. Adding a unit each time, we 
get FIVE, SIX, SEVEN, EIGHT, NINE. 

3. One, two, three, four, five, six, &ec., are called 
Numbers. 

A Number is, therefore, one unit or more. 

4, Arithmetic treats of numbers. 

5. Repeating the numbers in order—one, two, 
three, four, five, six, &c., is called Counting. 


QvEsTIoNS.—1. With what do we commence? What is one, a single 
thing, called ?—2. Of what is two made up? Of what is three madeup? If 
we go on, adding a unit each time, what do we get ?—3. What are one, two, 
three, four, &c., called? What isa Number ?—4. Of what does Arithmetic 
treat ?—5, What is Counting? Count nine. Count nine backwards—nine, 
eight, seven, &c. 


6 NOTATION. 


NOTATION. 


6. Every number has a name; as, one, two, three. 
In stead of writing out the name, however, we may 
represent it by a character; as, 1, 2, 3. 

Notation is the art of expressing numbers by 


characters. 
7. There are two systems of Notation, the Ar’abic 


and the Roman. 
° Ehe Arabic Notation. 


8, The Arabic Notation is so called because it 
was used by the Arabs. It employs these ten char- 
acters, called Figures :— 


NAUGHT 2 Two 6 sIx. 
Oo CIPHER 3 THREE 7 SEVEN 
ZERO 4 FOUR 8 EIGHT 
1 ONE 5 FIVE 9 NINE 


The first of these figures, 0, implies the absence 
of number. 0 cents means not a single cent. 

9. The greatest number that can be expressed 
with one figure is nine. All the numbers above 
nine are expressed by combining two or more 
figures. 

First, 1 is combined with each of the ten figures ; 
then 2, forming the twentzes ; then 8, forming the 
thirties ; then 4, forming the forties, &e. 


6. How may numbers be represented? What is Notation?—7. How 
many systems of notation are there? Name them.—8. Why is the Arabic 
Notation so called? How many characters does it use? What are they 
* called? Learn how to make the ten figures, and their names. What does 0 
imply?—9. What is the greatest number that can be expressed with one 
figure? How are all numbers above nine expressed ? 


THE ARABIC NOTATION. 


10, The numbers formed of two figures are 


10 
11 
12 
13 
14 
15 
16 
17 
18 

19 
20 
21 
29, 
23 
24. 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 


36- 


37 
38 
39 


ten 

eleven 
twelve 
thirteen 
fourteen 
fifteen 
sixteen 
seventeen 
eighteen 
nineteen 
twenty 
twenty-one 
twenty-two 
twenty-three 
twenty-four 
twenty-five 
twenty-six 
twenty-seven 
twenty-eight 
twenty-nine 
thirty 
thirty-one 
thirty-two 
thirty-three 
thirty-four 
thirty-five 
thirty-six 
thirty-seven 
thirty-eight 
thirty-nine 


40 
41 
42 
43 
44 
45 


55 
56 
57 
58 


EXEROISE. 


Count from 1 to 99. 


With what figure do the thirties all begin? 


forty 
forty-one 
forty-two 
forty-three 
forty-four 
forty-five 
forty-six 
forty-seven 
forty-eight 
forty-nine 
fifty 
fifty-one 
fifty-two 
fifty-three 
fifty-four 
fifty-five 
fifty-six 
fifty-seven 
fifty-eight 
fifty-nine 
sixty 
sixty-one 
sixty-two 
sixty-three 
sixty-four 
sixty-five 
sixty-six 
sixty-seven 
sixty-eight 
sixty-nine 


70 
71 
72 
73 
74 
75 
76 
17 
78 
79 
80 
81 
82 
83 
84 
85 
86 
87 
88 
89 
90 
91 
92 
93 
94 
95 
96 
97 
98 


99 


seventy 
seventy-one 
seventy-two 
seventy-three 
seventy-four 
seventy-five 
seventy-six 
seventy-seven 
seventy-eight 
seventy-nine 
eighty 
eighty-one 
eighty-two 
eighty-three 
eighty-four 
eighty-five 
eighty-six 
eighty-seven 
eighty-eight 
eighty-nine 
‘ninety 
ninety-one 
ninety-two 
ninety-three 
ninety-four 
ninety-five 
ninety-six 
ninety-seven 
ninety-eight 
ninety-nine 


Count from 99 to 1, backwards. 


The sixties? 


Write the following numbers in figures :—thirty-seven ; 
eleven; ninety-eight’; eighty-nine; twelve; twenty; five; 
fifteen; fifty. What system of notation have you just used? 


8 NOTATION. 


Unrrs, Tens, Hunpreps. 


11. Ten, we have seen, is expressed thus—10. 
Then 1 in the second place denotes one ten. So, 2 
in the second place (20) is two tens, &e. 

Any figure standing in the second place repre- 
sents so many tens. Hence it denotes ten times as 
much as if it stood in the first place. 

12, The first place is called the units’ place. The 
second is the tens’ place. 

13. Numbers greater than 99 must be expressed 
with more than two figures. A third place is thus 
required, which is called the hundreds’ place. 

A figure in the third place denotes ten times as 
much as if it stood in the second place, and a hun- 
dred times as much as if it stood in the first place. 

14. To express the even hundreds, place the sev- 
eral figures in the third place, with naughts after 
them. Thus :— 

100 one hundred 300 three hundred 
200 two hundred 400 four hundred, &c. 

15. Observe how the numbers between the even 

hundreds are expressed :-— 


101 one hundred and one 120 one hundred and twenty 
102 one hundred and two, &c. 121 one hundred and twenty-one 
110 one hundred and ten 201 two hundred and one 


111 one hundred and eleven, &c. | 202 two hundred and two, &e. 


11. How is tew expressed? What does 1 in the second place denote? 
2 in the second place? Any figure in the second place ?—12. What is the 
first place called? What is the second place called ?—13. How must num- 
bers greater than 99 be expressed? ‘What is the third place called? What 
value is denoted by a figure in the third place, compared with the second and 
the first ?—14. How are the even hundreds expressed ?—15, Learn how to ex- 
press numbers between the eyen hundreds, 


a 


THE ARABIO NOTATION. 9 


EXEROISE. 

Count from 800 to 400. Count backwards from 900 to 
800. Write on your slate the numbers from 500 to 600. 

Write 3 units, 6 tens, 9 hundreds (963); 9 units, 6 tens, 
8 hundreds; 7 tens, 5 units; 8 hundreds; 8 hundreds, 6 
units; five hundreds, one ten; two tens. , 

Express in figures three hundred and ninety-six. Two 
hundred and twelve. Eighty-one. Four hundred and two. 
Eight hundred and thirty. Six hundred. Seventeen. 


THOUSANDS. 


16. The greatest number that can be expressed with 
three figures, is 999. Next comes one thousand. 

One thousand is expressed thus, 1000—by put- 
ting 1 in the fourth place, which is called the 
thousands’ place. 

17, The number of thousands is shown by the 
figure in the fourth place. Thus :— 


2000 two thousand 6000 six thousand 

8000 three thousand 7000 seven thousand 
4000 four thousand 8000 eight thousand 
5000 five thousand 9000 nine thousand 


18. Ten thousand requires five figures to express 
it—10000. ‘The fifth place is called that of ten 
thousands. : 

19. A hundred thousand requires six figures to 
express it—100000. The sixth place is called that 
of hundred thousands. 


16. What is the greatest number that can be expressed with three 
figures? What comes next to 999? How is one thousand expressed? What 
is the fourth place called 2—17. By what is the number of thousands shown ? 
Give examples.—18. How many figures are required to express ten thousand ? 
What is the fifth place called ?—19. How many figures are required to ex- 
. press a hundred thousand? What is the sixth place called? 

1* 


10 NOTATION. 


20. We have now had six places named :—units, 
tens, hundreds, thousands, ten thousands, hundred 
thousands. 

These six places are divided into two Periods, of 
three figures each. The first Period is that of units; 
the second, that of thousands. 


n m2 ; 
cg ae a 
Ha = ® 
3 aa s Ps 
TOR OM wa ss n 
4 fa Bs = = Nn ~~ 
ae 1 r= 3 a a 
are) war) i> js) & i) 
a aoe ey 
Thousands Units 


21. The second period is that of thousands. To 
express a given number of thousands, write the 
number in the second period. If there are no 
figures for the units’ period, supply naughts. 

ExamMPLes.— Write four hundred and twenty-three thou- 
sand. To do this, write four hundred and twenty-three, as 


already shown—423—for the second period. Supply naughts 
for the units’ period, and we have the required number— 


423,000. 
So we write seventeen thousand, 17,000. 
Five hundred and one thousand, 501,000. 
Six hundred and twenty thousand, 620,000. 


If there are numbers corresponding to the places of the 
units’ period, set them there in stead of naughts. 
Forty-three thousand, two hundred and ninety, 43,290. 
Seven thousand, one hundred and five, 5! a ere 
One hundred thousand, and sixty-seven. (As 
there are no hundreds in the units’ period, supply 0.) 100,067. 


20. Name the first six places in order. How are these six places divided ? 
iat is the first period called? What is the second period called? Name 
the places of the first period. Name those of the second period.—21. How 
are we to express a given number of thousands? "What must be done, if 
there are no figures for the units’ period? Learn how to write the examples 
given. 


‘ 
Li 


THE ARABIC NOTATION. 11 


EXEROISE IN NOTATION. 
Write the following numbers in figures :— 
1. Five hundred and nine thousand. 
2. Sixty-three thousand, two hundred and seven. 
3. Eleven thousand, one hundred and eleven. 
. Seven hundred thousand and seventy. 
. Six thousand. Six hundred thousand. 
. Forty-three thousand and thirty-four. 
. Five hundred and twelve thousand, seven hundred. 
. Highty thousand, eight hundred and eight. 
. Nine hundred and ninety-nine thousand. 
10. Write the greatest number that can be expressed 
with three figures; with four figures; with five figures; 
with six figures. 


oot DO oO 


Mizutons, Brirtons. 


22. The third period is that of millions. 
It consists of three places,—millions, ten mil- 
lions, hundred millions. 


ExamMpPLes.—Two hundred million, 200,000,000. 
Four hundred and one million, 401,000,000. 
Seventy million, five hundred thousand, 70,500,000, 
Six million, seventeen thousand, and seven, 6,017,007. 


23. The fourth period is that of billions. 

It consists of three places,—billions, ten billions, 
hundred billions. | 

ExampLes.—One hundred and two billion, 102,000,000,000. 


Eleven billion, eleven thousand, and two, 11,000,011,002. 
Four billion, twenty million, and six, 4,020,000,006. 


22. What is the third period? Of how many places does it consist? 
Name them.—23, What is the fourth period? Of how many places does it 
consist? Name them. 


~ 


12 NOTATION. 


Summina Up. 


24, Name the periods in order, beginning at the 
right. | 


3 
m — 
m ro] 3 
S 3S wm m 
sj = mo Sale 
ae | m | 
as a og Sa ¢ om 
ma sa 2 Bree 
ee oe. uo] q uo) 
o 4 ww o = D5 5EO Fre ® 
mS a e) us) $e) = a rs uo) oe 
ee ee = ae =e - Bee =) Sl era 
bd Outs B.A - = | eriio ta 
Ce A | GS & MM oH 
uu HY San UU + 
4th Per. 3d Per. 2d Per. 1st Per. 
BILLIons MILLIONS THOUSANDS UNITS 


Name the places, beginning at the right. 


10 units make 1 ten. 
10 tens make 1 hundred. 

Hence, removing a figure one place to the right, 
diminishes its value ten times; removing it one 
place to the left, increases it ten times. 

25. Rute.— Write billions in the fourth period, 
millions in the third, thousands in the second, units 
in the first, filling the vacant places with naughts, 
so as to have three places in each period. 


EXEROISE IN NOTATION, 


Write the following numbers in figures, placing units 
under units, tens under tens, &c,:— 

1. Four hundred and seventy-one billion, six thousand. 

2. Ninety billion, three million, two thousand and four. 

8. Eight hundred million, sixty thousand, one hundred. 


24. What is the effect of removing a figure one place to the right? 
What is the effect of removing it one place to the left?—25. Give the rule 
for notation. is 


THE ARABIC NOTATION. 13 


4, Six hundred and forty thousand, one hundred and one. 
5. Nine million, fifty-seven thousand, and eight. 

6. Eleven billion, forty-one million, two hundred and ten, 
7. Thirty-six million, one hundred thousand, and twelve. 
8. Ten billion, ten million, ten thousand, and three. 

9. Six hundred and one million, two hundred thousand. 

10. Eighty-nine thousand, three hundred and nineteen. 

11. Twelve thousand, five hundred and eighty-seven. 

12. Four hundred billion, four million, forty thousand. 

13. Five million, eight hundred thousand, and seventy-five. 

14, One billion, ten million, two hundred thousand, and six. 

15. Fifty-seven million, three hundred and twenty-four. 

16. Four million, two hundred and seventeen thousand, 
and fifty-eight. 

17. Six hundred and nine billion, four hundred and sixty- 
six million, ninety-two thousand, three hundred and twenty- 
eight. 

The Roman Notation. 

26. The Roman Notation is so called because it 
was used by the ancient Romans. 

It employs seven letters. I. denotes one; V., 
five; X., ten; L., fifty; C., one hundred ; ah five 
iavaied M., one thetnands 

27. These TGthers are combined to express num- 
bers, according to the following principles :— 

1. If a letter is repeated, its value is repeated. 
XX. is twenty; ILI. is three. 

2. A letter of less value, placed after one of 
greater, unites its value to that of the latter. VI. 
1S SlX. 


26. Why is the Roman Notation so called? What does it use, to 
express numbers?--27. State the principles of the Roman Notation. 


14 NOTATION. 


8. A letter of less value, placed before one-of 
greater, takes its value from that of the latter. “IV. 
is four. | 

4, A letter of less value, placed between two of 
greater, takes its value from that of the other two 
united. LIV. is fifty-four. 

5. A bar over a letter increases its value a thou- 
sand times. V. is five thousand. 


TABLE. 
J. is One. L. is Fifty. 

II. “ Two. LX, “ Sixty: 

Tif. ‘“* Three. LXX. “ Seventy, 

LV¥2 }sKour: LXXX. * Eighty. 

Ves Bive: XC. “ Ninety. 

AN Mie Shh A C. * One hundred. 

VII. ‘* Seven. CI. “ One hund. and one. 
VIII. ‘ Eight. CIV. “ One hund. and four. 

IX. “ Nine. OX. “ One hund. and ten. 

Xu4 Ten: CC. “ Two hundred. 

XI. ‘ Eleven. CCC. * Three hundred. 
XII. ** Twelve. CCCC. “ Four hundred. 
XIII. “ Thirteen. D. “ Five hundred. 
XIV. “ Fourteen. DC. * Six hundred. 

XV. “ Fifteen. DCC. ** Seven hundred. 
XVI. “ Sixteen. DCCC. “ Eight hundred. 
XVII. “ Seventeen. DCCCO. * Nine hundred. 

XVIII. ‘‘ Highteen. M. “ One thousand. 

XIX. “ Nineteen. MM. “ Two thousand. 

XX. ‘ Twenty. MMM. “ Three thousand. 
XXI. ‘* Twenty-one. MMMM. “ Four thousand. 
XXX. “ Thirty. V. “ Five thousand. 

XL. “ Forty. X. “ Ten thousand. 


What is the effect of placing a bar over a letter? How is five thousand 
denoted? Learn the Table. 


a ae 


28. We may, then, express numbers in three 


ways :— 
1. With words, as is usual in printed books. 
2. With jigures, by the Arabic Notation, as in 
accounts and calculations. 
3. With letters, by the Roman Notation, as in the 
headings of chapters. 


EXERCISE IN NOTATION. 15 


EXERCISE IN NOTATION. 


Write the following numbers first by the Arabic, and then 
by the Roman, Notation :— 


A. 
2. 


4. 
5. 


Twelve. 6. One thousand. 
Fifty-seven. 7. Ninety-nine. 

. Nine hundred. 8. Seven hundred. 
Eighty-six. 9. Sixty-two. 
Nineteen. 10. Four thousand. 


‘ 11. Five thousand six hundred and seventy-three. 
12. Three hundred and seventy-two. 
18. Two thousand eight hundred and forty-one. 
14. Nine thousand and twenty-seven. 
15. Fifteen hundred and thirty-five. 


Express the following numbers according to the Roman 
Notation: 12; 1,000; 749; 18; 208; 96; 660; 488; 29; 


2,040; 85; 555; 10,801; 79; 5,00 


8,186; 119. 


2: 87; 894; 999; 2.062; 


‘Express the following numbers according to the Arabic 
Notation: XIJ. LI. VIII. XLII]. XVI. LXXXIX. XCVIII. 
COI. DXX. XXXIV. MD. IX. MCCXV. DCCOVII. XIV: 
MDCLXVI. V. 


evil 


lv. eciv. xxxili, xix. xlviii. 


XC. Cxxi. xv. Ixii. 


28. How many ways are there Be expressing numbers? What are they, 
and where is each used? 


2 


a 


16 NUMERATION. 


NUMERATION. 


29. Numeration is the art of reading numbers ex- 
pressed by figures. 


30. In reading numbers, the following principles 
apply :— 

1. We read by periods. Hence, if there are more 
than three figures, point off the number into periods 
of three figures each, beginning at the right. 

2. Always begin to read at the left. 

3. The right- hand figure and the right-hand 
peaod are never named as units, the word wnits 
being understood. We read 7 as seven, not seven 
units ; 400 i is read four hundred, not four hundred 
units. 

4. Places containing 0 must be passed over in 
reading. We read 1062 one thousand and siaty- 
two, not one thousand, no hundred, and sixty-two. 


31. Rute.— Beginning at the right, point off the 
number into periods of three figures each. 

Beginning at the left, read the figures im cach 
period as if they stood alone, adding the name of 
the period m every case except the last. 


EXAMPLES. 


10,709 Ten thousand, seven hundred and nine. 
401,840 Four hundred and one thousand, eight hundred and forty. 
6,023,070 Six million, twenty-three thousand, and seventy. 


29. What is Numeration?—30. How do we read numbers? If there are 
more than three figures, what do we do? At which side do we begin to 
read? What is said of the right-hand figure and the right-hand period? 
What must be done in the case of Heo cont ns 0?—81, Give the rule 
for Numeration. 


EXERCISE IN NUMERATION. IZ 


42,110,000 Forty-two million, one hundred and ten thousand. 
870,025,002 Hight hundred and seventy million, twenty-five 
thousand, and two. 
1,001,000,011 One billion, one million, and eleven. 
19,056,007,000 Nineteen billion, fifty-six million, seven thousand. 
123,400,789,000 One hundred and twenty-three billion, four hundred 
million, seven hundred and eighty-nine thousand. 


1100 is read one thousand one hundred, or eleven hundred. 
1200 “* “ one thousand two hundred, or twelve hundred, &c. 
EXERCISE IN NUMERATION. 


Read the following numbers :— 


is 903 | 15. 87123645 | 29. MDCCCLXIII. 
2. 8600 | 16. 476674983429 | 30. VOCOXCIX. 
3. 91 | 17. 82000000117 | 31. DCLXXXV. 
4, 100075 | 18. 1413 | 32 XXXVIIL 
5B. 98282] 19. 103600028 | 338. MDXIX. 
6. 8800000 | 20. 50500005050 | 84. DOCXVIL. 
%, 463925 | 21. 442671376000 | 35. MCXI. 
- 8. 1650 | 22. 15000027 | 36. LIX. 
9. 1040400 | 23. 998899989898 | 37. DXCVI. 
10. 26308 | 24, 203013310031 | 38, LAL. 
11. 8005042 | 25. 410° }°89; . / XCOXX VIL 
12. 741607 | 26, £418760 | 40. TeX: 
13. 8821060 | 27. 83227 | 41. MDOXVIII. 


14: 600007 | 28. 14603000 | 42. CCCCXLIYV. 


Review Questions.—What is a Unit? Whatisa Number? Of 
what does Arithmetic treat? What is Counting? What is Notation? 
Name the two systems of notation. What characters are used in the 
Arabic Notation? Name the periods in order, beginning at the right. 
Name the places. Give the rule for expressing numbers in figures. 
What characters are used in the Roman Notation? State the princi- 
ples on which they are combined. What are used to express num- 
bers, in making calculations? What, in accounts? What, in head- 
ings of chapters? What is Numeration? Give the rule for reading _ 
numbers. 


s 


18 ADDITION. 


ADDITION. 


32. Two men are riding and three are walking; 
how many men are there in all? 

Here we are required to find one number containing as many 
units as 2 and 3 together. This process is called Addition. 

33. Addition is the process of uniting two or 
more numbers in one. 

The one number thus obtained is called the Sum, 
2 and 3 are 5; 5 is the sum. 


Appition TABLE. 
0 and 1 are 1; 0 and 2 are 2; 0 and any number make that number. 


1 and 2 and 8 and 4 and 5 and 
l are 2 l are 3 lare 4 l are 5 l are 6 
2 are 3 2are 4 2are 5 2 are 6 2 are 7 
8 are 4 3 are 5 3 are 6 38 are 7 8 are 8 
4 are 5 4 are 6 4 are 77 4 are 8 4 are 9 
5 are 6 5 are 7 5 are 8 5 are 9 5 are 10 
6 are 7 6 are 8 6 are- 9 6 are 10 6 are 11 
Tare 8 T are 9 "7 are 10 7 are 11 7 are T2 
8 are 9 8 are 10 8 are 11 8 are 12 8 are 13 
9 are 10 9 are 1l 9 are 12 9 are 138 9 are 14 

10 are 11 | 10 are 12 | 10 are 13 | 10 are 14 | 10 are 15 

6 and 7 and 8 and - 9 and 10 and 
lare 7 lare 8 l are 9 1 are 10 1 are 11 
2 are 8 2 are 9 2 are 10 2 are ll 2 are 12 
3 are 9 3 are 10 8 are 11 8 are 12 8 are 138 
4 are 10 4 are 11 4 are 12 4 are 13 4 are 14 
5 are 11 5.are 12 5 are 13 5 are 14 5 are 15 
6 are 12 | 6 are 13 | 6 are 14 | 6 are 15 | 6 are 16 
7 are 13 7 are 14 7 are 15 7 are 16 7 are 17 
8 are 14 8 are 15 8 are 16 8 are 17 8 are 18 
9 are 15 9 are 16 9 are 17 9 are 18 9 are 19 

10 are 16 |,10 are 17 | 10 are 18 | 10 are 19 | 10 are 20 


ADDITION. 19 


It is necessary to know the Tables perfectly, so as to say them 
backwards or forwards, out of order as mel} asin order. They must 
be mastered before going on. 

34, Addition is denoted by an erect cross +, 
called Plus, placed between the numbers to be 
added. 6+ 5 is read six plus jive, and means that 
siz and five are to be added. 

35. T'wo short horizontal lines =, placed between 
two quantities or sets of quantities, denote that they 
are equal. 6+5=11 is read siz plus five equals 
eleven, and means that the sum of six and fie is 
eleven.. 

36. Observe that if 


3+2=5 4+5=—9 8+7=—10 5+8=13 
then then then then 

134+2=15 344+5=39 53+7=60 25+8=33 

934+9=95 4445=49 68+7="70 454+8=53 


834+2=35, &c.| 544+5=59, &c.| 73+7=80, &.| 85+8=—93, &e. 


37. Observe that 4+5=9 and 5+4=9. 
Hence, when numbers are to be added, it makes 
no difference which is taken first. 


EXEROISE ON THE ADDITION TABLE. 


How many are 5 and4? 4and5? 24and5? 25 and 4? 
4and55? 94and5? 15and4? 8and2? 2 and 73? 

How many are 7 and 1? land 7? 67 and1? 7 and 81? 
lland7? 3and6? 6and38? 46and3? 6 and 23? 


82. In the example given, what are we required to find? What is this 
process called ?7—33. What is Addition? What is the result of addition 
called? Recite the Table.—34 By what is addition denoted? What does 
6+5 mean?—35. Describe the sign that denotes equality. What does 6+5 
=11 mean ?—36 If 3+2=5, then what follows? How much is 5+8?% What 
follows?—37. How much is 4+5? Howmuch is 5+4? What principle is 
laid down respecting numbers to be added? 


20 ADDITION. 


How many are 6 and2? 66 and2? 2and86? 82 and6? 
2and6? 12and6? 382and6? 6 and 52? 16 apd 2? 
How many are 8 and2? 2 and 8? YVand38? 8 and 7? 
18 and 2? 97 and 8?. 82*and 8? 78 and2? 7 and 53? 
How many are 3 and 5? 55 and3? 38and95? 23 and 
5? 18 andd5? 4and2? 64and2? 62 and4? 4 and 72? 
How many are 5 and5? 2and7? 45 and 5? 52 and 7? 
Wand 7? 387 and2? 2and5? 42 and5? 4 and 6? 
How many are 4 and 8? 84 and 3? 98 and 4? 14 and 3? 
4Aand4? 84and4? 4and 54? 9and8? 4 and 9? 
Find the sum of 1, 2, and 3. 8+248,. 44+1+5. 64341. 
8+44+2. 64343. 44647. 347410. 2043844. 
Count by twos, commencing 2, 4, 6, 8, &c., up to 100. 
Count by threes, commencing 8, 6, 9, 12, &c., up to 99. 
Count by fours, commencing 4, 8, 12, 16, &c., to 100. 
Count by fives, commencing 5, 10, 15, 20, &c., to 100. 


MENTAL EXEROISES. 


1. Seven metals were known to the ancients; 43 have been 
discovered since. How many metals are now known? 

Ans. 7+48 metals, or 50 metals. 

2. How much will a boy earn in two weeks, if he earns 
5 dollars the first week and 2 the second? 

3. John Adams was president four years. He was 61 
when he entered on the duties of the office; how old was he 
when he left it? 

4, A gardener set out 9 trees one day, and 8 the next; 
how many did he set out both days? 

5. If a house has 8 windows on one side, 6 on another, and 
4 on a third, how many windows has it in all? ) 

6. Napoleon had 4 brothers and 3 sisters, besides 5 that 
died in infancy; how many brothers and sisters had he in all? 


EXERCISES IN ADDITION. 91 


7. How many pounds will a pair of chickens weigh, if 
each weighs three pounds? 

8. My house is at the north end of a lake; Mr. A’s is 8 
miles south of the south end. Ifthe lake is 5 miles long, how 
far is it from my house to Mr. A’s? 

9. A farmer who has 7 cows, buys 6 more; how many has 
he then? 

10. In a jar that weighs 6 pounds, I put 11 pounds of but- 
ter; how much will the whole weigh? 

11. A boy who bought a quire of paper for 20 cents, sold 
it so as to gain 5 cents; how much did he sell it for? 

12. Washington was born in 1732. George III. was born 
6 years later; what was the date of his birth? 

13. A steamboat starts with 72 passengers. Three miles 
down the river, it receives 7 more passengers. How many 
has it then? : 

14. The Earth has 1 moon; Jupiter has 4 moons; Saturn, 
8; Uranus, 6; Neptune, 1; how many moons does that make 
altogether 2 


38. IPrimciples of Addition. 


1. We must add things of the same kind. There- 
fore, in setting down numbers to be added, place 
_ units under units, tens under tens, &c. 

2. The sum of units is units; of tens, tens; &c. 

3. Always begin to add at the right. 

4, Find the sum of each column; and, if it is ex- 
pressed by one figure, set it down under the column 
added. 

38. How must we set down numbers to be added? Whyso? What is 


the sum of units? Of tens? Of hundreds? Where must we begin in add- 
ing? If the sum of each column is expressed by one figure, where must we 


set it? t he 


22 ADDITION. 


Exampir.—Add four million; three hundred and twenty- 
six thousand, two hundred and forty-seven ; fifty-three thou- 
sand, four hundred and ten; and two hundred and twenty- 
one. 

Operation.— Write down the numbers, units under units, 
tens under tens, &c. 

Begin to add at the right. 

1st column. 1 and 7 are 8. Set down 8. 


9d. 2 and 1 are 3, and4is 7. Set down 7. 4000000 
3d. 2 and 4 are 6, and 2 is 8. Set down 8. 326247 
4th. 8 and 6 are 9. Set down 9. 53410 
5th. 5 and2 are 7. Set down 7. 221 


6th. Bring down 8. 7th col. Bring down 4. Ans. 4379878 


‘Proof of Addition. 


39, Proving an example is finding whether the 
work is correct. 

49. Addition is proved by adding the columns 
from the top downward. If the sum is the same as 
when they are added from the bottom upward, we 
infer that the sum is right. 


This Proof is based on the fact that, when numbers are 
to be added, it makes no difference in what order they are 
taken. The sum will be the same. If an error has been 
made in adding up, it is not likely to be repeated in adding 
down, and will thus be detected. Be 

Exampie.—Prove the above example. Add 4000000 
each column from the top downward. 7 and 1 326247 
are 8. 4and1are5, and 2is7. 2and4are6, 58410 
and 2 is 8. 6 and 3 are 9. 2 and 5 are 7%. 221 
Bring down 3. Bring down 4. Answer, 4879878 ~~~ 
—the same as before. Hence the work is right. so 


Apply these principles in the example given.—89. What is meant by 
Proving an example ?—40. How is addition proved? On what is this proof 
based? Prove the example just eigen. 


’ EXERCISE IN ADDITION. 23 


EXAMPLES FOR THE SLATE. 


41, Read the numbers added. Prove each example. 
. Add 600128, 154235, 34400, and 221. Ans, 788979. 
. Add 85026371, 41005, and 1810000. Ans. 86877376. 
. Find the value of 12344455111 4+ 20234412120, 
. Add 297661031851, 1135204022, 3115, and 1520010. 
. What is the sum of 2014100 +2288 + 364114+ 322201 ? 
6. Find the sum of thirty-one; one hundred and eleven; 
twenty thousand, four hundred and forty-two; seventeen 
‘thousand, one hundred and eleven; and sixty thousand, one 
hundred and three. Ans. 97798. 
7, Add together seventeen million; one hundred and fifty 
thousand, one hundred; eight hundred and nine thousand, 
two hundred and seventy-two; and forty thousand, three 
hundred and sixteen. Ans. 17999688. 
8. What is the sum of four million, eight hundred and 
twelve thousand, one hundred and two; thirty-one thousand, 
six hundred and twenty; one million, one thousand, and 
forty-five ; and three thousand and thirty ? Ans. 5847797. 
9. Add together three hundred and twenty; one hundred 
and eleven million, two hundred and twelve; forty thousand, 
one hundred and thirty-two; and two million, one hundred 
and thirty-seven thousand. Ans. 113177664. 
10. Find the sum of twenty billion, one thousand, and one; 
four thousand and eleven; four hundred and seven million, 
twenty thousand, six hundred and forty-two; and sixty-three 
thousand, one hundred and three. Ans. 20407088757. 
11. A merchant sells $10000* worth of goods one day; 
$5123 worth, the next; and $2486 worth, the next. How 
much does he sell in all? 


or HH OF WO eH 


* This mark ($) denotes dollars. It is always placed before the num- 
ber. $1000 is read @ thousand dollars, 


94 ADDITION. 


12. Three score are sixty. How many are three score and 
ten? 

13. If I travel 1246 miles by steamer, 732 by railroad, and 
21 by stage, how far do I travel altogether? 

14, An army contains 23022 infantry privates; 710 in- 
fantry officers; 4000 cavalry, including officers; and 164 mu- 
sicians. How many men in all in the army? 

15. A. sells a vessel for $15420, which is $1355 less than 
it cost. What did the vessel cost ? 

16. The estate of a deceased man was divided as follows: 
his wife received two hundred and tén thousand dollars; his 
daughter, forty thousand two hundred and fifty dollars; his 
elder son, twenty-one thousand five hundred and six dollars; 
his younger son, twenty thousand one hundred and forty-two 
dollars. What was the value of the estate? Ans. $291898. 

17. The planet Mars is 145,205,000 miles from the sun. 
Jupiter is 350,610,500 miles farther. How far is Jupiter? 


Carrying. 

42, The sum of a column may make more than 
one figure. 

Exampie.—Add 487 and 975. A87 

Begin at the right. 5 and 7 are 12—2 975 
units and 1 ten. Set down 2 in the units’ 1469 
place, and add the 1 ten to the other tens. 

1 and 7 are 8, and 8 is 16. 16 tens are 6 tens 
and 1 hundred. Set down 6 in the tens’ place, and 
add the 1 hundred to the other hundreds. . 

1 and 9 are 10, and 4 is 14—14 hundreds, * 
hundreds and 1 thousand. Answer, 1462. 


42. With the given example, show what is meant by carrying. Give 
the rule for carrying. 


CARRYING IN ADDITION. 25 


43, This adding of the left-hand figure is called 
Carrying. 

44, Rute ror Carryine.— When the sum of a 
column ws over 9, set down the right-hand figure, 
and carry the left-hand figure or figures to the next 
column. 

EXAMPLES FOR THE SLATE. 


45, Read and add the following numbers. ‘Prove each 
example. 


(1) (2) (8) 
24897 43345678 123423434559 
64. 1123355 23785432977 
234567 7893 — 9876543696 
2357911 54689 - 51002789 
34567890 734321 10200859 


4, Add 123405, 54210, 1794322, and 6541. Ans. 1978478. 

5, Add 4275602, 45706, 5567801, and 365. Ans. 9889474. 

6. Add 23, 6794, 896423, 597, and 16019. Ans. 919856. 

7. What is the value of 965482190006 + 4063 + 8127299837 
+ 102009 + 9238675 + 67. Ans. 973618834657. 

8. Find the number of days in a year, there being 31 days 
in January, 28 in February, 81 in March, 30 in April, 31 in 
May, 30 in June, 31 in July, 31 in August, 30 in September, 
31 in October, 80 in November, and 31 in December. 

9. The amount of treasure exported from California in 
1861 was $40639089. This was $1664256 less than in 1860. 
What was it in 1860? 

10. The two largest cities in Europe are London and Paris. 
The population of London in 1851 was 2362236; that of Paris, 
1058262. What was the population of both? 

11. Benjamin Franklin was born in 1706, and died at the 
age of ae In what year did he die? 


26 ADDITION. 


12. Rhode Island, the smallest state in the Union, contains 
1306 square miles. Texas, the largest state, contains 236198 
square miles more than Rhode Island. How many square 
miles in Texas ? 

18. In 1850, South Carolina produced 800901 bales of cot- 
ton; Alabama, 564429. How much did both produce? 

14, The United States is made up of the Atlantic Slope, 
which contains 967576 square miles; the Mississippi Valley, 
which contains 1237311 square miles; and the Pacific Slope, 
which contains 778266 square miles.) How many square 
miles in the whole United States? 

15. In 1850, 4203064 copies of papers and periodicals were 
printed in Maine; 8067552, in New Hampshire; 2567662, in 
Vermont; 64820564, in Massachusetts; 2756950, in Rhode 
Island; 4267932, in Connecticut. How many were printed 
in all the New England states? Ans. 81683724. 


46. Rule for Addition. 


1. Set units under units, tens under tens, de. 

2. Beginning at the right, find the sum of each 
column. 

3. Lf the sum is expressed by one figure, write 
it under the column added; if not, set down the 
right-hand figure, and carry the left-hand figure or 
jigures to the next column. 


EXEROISE. 
44, The following examples are to be practised until they 
- can be added at sight up and down, naming the results only. 
Thus in Example 1 :—jive, eleven, thirteen, fifteen, | 
twenty-one, thirty, thirty-seven—set down 7, and carry 3. 
Three, seven, fourteen, &e. 


@) 
899697 
125429 
4898438 
974583 
583162 
236972 
119876 


612345 


(5) 
958576 
328492 
223967 
225523 
221679 
120739 
653865 
862781 
426893 


4022515 


EXERCISES IN ADDITION. 


(2) 


(3) 


(9) 
838725889 
189404376 
397846748 
578757467 
949549259 
499699451 
887415762 
7393538343 
568239774 
6 
4 


28135 
16296 
906147 


6814842647 


891985 342687 
263882 165431 
885487 267994 
298688 319416 
656599 948668 
6774.94. 275773 
681395 139915 
684923 289076 
(6) (7) 
758318 669786 
272638 359628 
364773 694279 
525822 946335 
294987 834569 
162856 179145 
175902 5T9TST 
943655 954852 
682354 864351 
4181305 6082702 
(10) . 
6158874991 
8254889782 
965449873 
48334387997 
2511486766 
1896479693 
65769958 
4747786854 
4598748765 
2152759921 
5439765653 
8901729836 
50527180089 


27 


(4) 
185232 
965368 
905596 

78748 
588933 
879370 
899477 
276431 


(8) 
895939 
765838 
678930 
514831 
455922 
379828 
263955 
675869 
416970 


5048082 


(11) 
652824777 
863928996 
896956866 
885999586 
992845696 
894896986 
885699577 
864712998 
893888579 
878994886 
893989466 
784985886 


10389724299 


98 SUBTRACTION. 


SUBTRACTION. 


48. Threc boys are on the lawn. Two go into 
the house; how many are left ? 

Here we are required to take 2 from 3, or to find the dif- 
ference between 2 and 3. This process is called Subtraction. 

49, Subtraction is the process of taking one num- 
ber from another. 

50. The smaller number must always be taken 
from the greater. We can take 2 from 3, but not 
3 from 2. 

SuBTRACTION TAxxe. 


0 from 1 leaves 1; 0 from 2 leaves 2; 0 from any number leaves 
that number. 


1 from 2 from 3 from 4 from 5 from 
lleaves 0 | 2leavesO | S8leaves0| 4 leaves 0 | 5 leaves 0 
2leaves 1] 3leaves1} 4leaves1]} 5 leaves1!' 6 leaves 1 
3 leaves 2| 4leaves2]| 5 leaves2| 6 leaves 2 | 7 leaves 2 
4 leaves 3 | 5 leaves3| 6leaves8| 7 leaves3j| 8 leaves 3 
5 leaves 4 | 6leaves4] 7 leaves4} 8 leaves4)| 9 leaves 4 
6 leaves 5 | Vleaves5 | 8 leaves5 | 9 leaves 5 | 10 leaves 5 
4 leaves6 | 8 leaves6 | 9 leaves 6 | 10 leaves 6 | 11 leaves 6 
8 leaves 7 | 9 leaves 7 | 10 leaves 7 | 11 leaves 7 | 12 leaves 7 
9 leaves 8 | 10 leaves 8 | 11 leaves 8 | 12 leaves 8 | 18 leaves 8 

10 leaves 9 | 11 leaves 9 | 12 leaves 9 | 13 leaves 9 | 14 leaves 9 


ee een En 


6 from 7 from 8 from 9 from 10 from 
6 leaves 0 | Tleaves0 | 8 leaves 0} 9 leaves 0 | 10 leaves 0 
"leaves 1| 8leaves1| 9 leaves 1 | 10 leaves 1 | 11 leaves 1 
8 leaves 2 | 9 leaves 2 | 10 leaves 2 | 11 leaves 2 | 12 leaves 2 
9 leaves 3 | 10 leaves 3 | 11 leaves 8 | 12 leaves 3 | 18 leaves 8 
10 leaves 4 | 11 leaves 4 | 12 leaves 4 | 13 leaves 4 | 14 leaves 4 
11 leaves 5 | 12 leaves 5 | 13 leaves 5 | 14 leaves 5 | 15 leaves 5 
12 leaves 6 | 18 leaves 6 | 14 leaves 6 | 15 leaves 6 | 16 lea 
13 leaves 7 | 14 leaves 7 | 15 leaves 7 | 16 leaves 7 | 17 lea 
14 leaves 8 | 15 leaves 8 | 16 leaves 8 | 17 leaves 8 | 18 leaves 8 
16 leaves 9 | 17 leaves 9 18 leaves 9 | 19 leaves 9 


15 leaves 9 


SUBTRACTION. 99 


51. The number to be subtracted, is called the 
Subtrahend. That from which the subtrahend is to 
be taken, is called the Minuend. The result, or what 
is left, is called the Remainder. 

2 from 3 leaves 1; 2 is the subtrahend, 3 the 
minuend, 1 the remainder. 

52. Subtraction is denoted by a short horizontal 
line =, called Minus, placed before the subtrahend. 

8—2 is read three minus two, and means that 2 
is to be subtracted from 3. 

53. Observe that if 


3—2=—1 7—3=—4 7—3=—4 11—10=1 
then then _ then then 

18—2=11 47—3=—44 27—23—=4 31—10=21 

23—2=21 57—3—54 T7—73—4 51—10=41 


33—2=—81, &c.| 67—3=—64, &c.| 87 -—83=4, &e.| 91—10=81, Xe. 


EXEROISE ON THE SUBTRAOTION TABLE. 


How many does 4 from 6 leave? 4 from 16? 4 from 36? 
4 from 562 14from16? 74from 76? 24 from 26? 

How much is 9-38? 29-8? 49-8? 69-3? 89-3? 
9—6? 89—6? 59—6? 79-6? 99—6? 6-1? 11-1? 

Take 2 from 4. 2 from 54. 2 from 94. 2 from 24. 8 
from 4. 3 from 64. 8 from 34. 4 from 4. 4 from 94, 

How much is 8—5? 78—5?°18—5? 48-5? 8—3? 28 ~~ 
—3? 88-8? 68-3? 68—5? 5—5? 25-5? 35-5? 

Subtract 3 from 5. 3 from 55. 83 from 35. 48 from 45. 
2from5. 2from15. 12 from15. 22 from 25. 2 from 85. 

48. In the example given, what are we required to do? What is this 
i) 33 calied ?—49. What is Subtraction?—50. Which is the number to be 
; oo Recite the Table. What does 0 from 1 leave? 0 from 2? 
0 any number?—5l. What is the number to be subtracted called? 
What is the number from which the subtrahend is to be taken called? 


What is the result called ?—52, How is subtraction denoted ?—53. How much 
is 3—2? What follows? How much is 11—10? - What follows? 


30 SUBTRACTION. 


How many does 5 from 7 leave? 5 from 57? 5 from 87? 
2from 7? 2from17? 2from67? 4from7? 4 from 27? 
How much is 8—4? 18—4? 88-84? 9-2? 69-2? 
89-82? 49-42? 9-7? 19-172? 9-4? 59-4? 
Take 9 from 10. 9 from 40. 9 from 60. 8from10. 8 
from 16. S8from17. Y%fromi1l. 6 from12. 5 from 10. 
Count backward by twos from 100. Thus: 100, 98, &. 
Count backward by fives from 100. Thus: 100, 95, &c. 
Count backward by tens from 100. Thus: 100, 90, &c. 
Count backward by twos from 99. Thus: 99, 97, &c. 


MENTAL EXEROISES. 


1. Fifty metals are now known. Seven were known to 
the ancients. How many have been discovered since? 

Ans. 50—7 metals, or 43 metals. 

2. A boy buys a paper for 5 cents. He gives the news- 
man 10 cents. How much change will he get? 

8. Ellen is 14 years old, and Jane is 6 years younger. 
How old is Jane? 

4, A farmer who has 15 sheep, sells 7 of them. How 
many has he left? 

5. Aman is on his way home from a town twelve miles 
distant. When he has walked nine miles, how much farther 
has he to go? : 

6. A flower-girl who starts with 24 nosegays, comes back 
with 4. How many has she sold? 

7. Five out of nine eggs turned out bad; how many were 
good? 

8. If I buy a cow for $29, and sell her for $23, how h 
do I lose? | 

9. Leaving home with $15, I spend $6 and give $3 away. 
How much have I left? 


PRINCIPLES AND PROOF. 31 


54. Principles of Subtraction. 


1. The smaller number is the one to be sub- 
tracted. Set it under the greater. 

2. As we must subtract things of the same kind, 
place units under units, tens under tens, &e. 

3. The difference between units and units is units; 
between tens and tens, tens; &c. 

4. Begin to subtract at the right. 

5. Take each figure of the subtrahend from the 
one above it, and set the remainder under the figure 
subtracted. 

ExampitE.—From eight million six hundred and forty 
thousand nine hundred and fifty-seven, subtract two hundred 
and ten thousand four hundred and thirty-six. 

Operation.—. Write the smaller number under the greater, 
units under units, tens under tens, &c. Begin at the right. 

6 from 7 leaves 1; set it down under the 6.. 


8 from 5 leaves 2. 4 from 9 leaves 5. 0 from sed 
O leaves 0. 1from4leaves 3. 2 from 6 leaves piste Sele, 
4, Bring down 8 Answer, 8430521. 8430521 


Proof of Subtraction. 


55. Add the remainder and subtrahend. If their 
sum is equal to the minuend, the work is right. 


Exampte.—Prove the example just 
given. Add the remainder to the sub- Rem. 8430521 
trahend. Their sum is 8640957, which % 210436 
is equal to the minuend. Therefore the wn 8640957 
work is right. 


54. How are the numbers to be set down in subtraction? Of what de- 

ination is the difference between units and units? Between tens and 
tens? Where must we begin to subtract? How are we to proceed? Solve 
the example given.—55. How is subtraction proved? Prove the example just 
given, 


o2 SUBTRACTION. 


EXAMPLES FOR THE SLATE. 


56. Read the numbers given and the remainders. Prove 
each example. 
1. From 1908647 take 2321. 6. 4678759—2678657. 
2. From 897628 take 84527. 7. 579583099—20052064, 
8. Take 1481 from 11463845, 8. 78199087— 60152085. 
4, From 673485 take 204382. i 3976189653 — 1042053, 
5. Take 38148 from 1159463. | 10. 12186947285—103014. 
11. From six hundred and fifty-seven thousand eight hun- 
dred and forty-nine, subtract five hundred and twenty-one 
thousand six hundred and sixteen. Ans. 136233. 
12. Take four million seven thousand one hundred and 
thirty-one, from forty-six million eighty-nine thousand five 
hundred and forty-two. Ans, 42082411. 
18. The subtrahend is three million four hundred and 
forty-three thousand. The minuend is five million eight 
hundred and forty-nine thousand and six. What is the re- 
mainder ? Ans, 2406006. 
14. From eight hundred and forty-eight billion six hun- 
dred and ninety-seven million and ten, take thirty-one billion 
one hundred and forty-two million. Ans. 817555000010. 
15. How much more is twelve billion eight hundred and 
seventy-nine million three hundred and sixty-four, than one 
billion two hundred and thirty-five million and twenty-two? 
Ans. 11644000342. 


Carrying. 


57, The lower figure may be greater than ais 
one above it. 

Exampie.—From 738 take 419. 

Begin at the right. We can not take 9 me 
from 8, because 9 is greater than 8, — 


CARRYING. 33 


Hence from the 3 tens we take 1 away, 2 
leaving 2 tens. The 1 ten taken away is is 
equal to 10 units, which we add to the 8 —— 
units, making 18. ae 

Now we can subtract 9. 9 from’18 leaves 9; 
set it down. 1 from 2 (not 3) leaves 1; 4 from 7 
leaves 3. Answer, 319. 

To balance the 10 units added to the 8, we took 
away 1 of the tens from the upper line. But in 
stead of diminishing the upper figure 1, we may 
add 1 to the figure below it. This gives the same 
result, and, being more convenient, is the mode 


generally pursued. . 738 
Thus: 9 from 18, 9. 1 and 1 are 2; 2 from 8 419 
leaves 1. 4 from 7,3, Answer, 319. 319 


58, This adding of 1 to the lower figure is called 
Carrying. 

59. We may have to carry several times in suc- 
cession. | 

ExamPpie.—From 10000 take 9999. 9 from10 ~~ 10000 
leaves 1; set it down and carry 1. 1 and 9 are 9999 
10; 10 from 10 leaves nothing. Carry1; land9  _*°** 
are 10; 10 from 10 leaves nothing. Oarry 1; 1and 1 
9 are 10; 10 from 10 leaves nothing. Answer, 1. 

60. Rote ror Carryina.— When the lower fig- 
ure is greater than the one above it, add 10 to the 
upper figure, subtract, and carry 1 to the next lower 


ti 
57. Show, with the example given, how we proceed when the lower fig- 


ure is greater than the one above it.—58, What is this adding of 1 to the 
lower figure called?—59. Show how we miay have to carry several times in 
succession.—60. Give the rule for carrying: 


9* 


34 SUBTRACTION. 


EXAMPLES FOR THE SLATE. 


61. Read the numbers. Subtract. Prove each example. 


(1) (2) (8) 


6310475 129060418801 600173240 
3501768 68431160424 25823186 
(4) (5) (6) 
19372848 801009647258 415673285 
9445276 77718194386 6906356 ~ 
7. Take four thousand and thirteen, from one million and 
eleven. Ans. 995998. 
8. From sixty-five thousand and seven, subtract nine hun- 
dred and ninety-nine. Ans. 64008. 


9. From seven hundred and thirty-tour thousand five 
hundred and twenty-one, subtract eighty-four thousand two 
hundred and eighty-three. Ans. 650288. 

10. Subtract seven hundred and sixty-two million nine 
hundred thousand and seventy, from one billion fourteen 


thousand and nine. Ans. 237118989. 
11. From two hundred and fifty-five thousand, take seven 
hundred and two. Ans. 254298. 


62. Rule for Subtraction. 


1. Set the smaller number under the greater, units 
under units, tens under tens, ke. 

2. Beginning at the right, take each figure of the 
subtrahend from the one above at, and set the re- 
“mainder under the figure subtracted. 

3. If the lower figure vs greater than the one 
above it, add 10 to the upper jigure, subtract, and 
carry 1 to the next lower figure. 


EXAMPLES FOR PRACTICR. 35 


When a whole is given and one of its parts, what is the 
rule for finding the other part? 


Subtract the given part from the whole. 
When a whole is given and all its parts but one, what is 
_ the rule for finding that one? 

Add the given parts, and subtract their sum from 
the whole. 

When the year of a person’s birth and that of his death are 
given, what is the rule for finding his age? 

Subtract the earlier date from the later. 


EXAMPLES FOR THE SLATE. 


1. Howard, the philanthropist, was born in the year 1726. 
He died in 1790. To what age did he live? 

2. Hudson explored the river called by his name in 1609. 
How long was this after the discovery of America, which took 
place in 1492? 

3. How many years have elapsed from the ‘dleontacs of 
America to the present time? 

4, A man worth $47650, leaves $29855 to his wife and 
the rest to his son. What is the son’s portion? 

5. At an election 12572 votes were cast, of which the suc- 
cessful candidate received 7698. How many votes did the 
other candidate receive ? 

6. In the above election, what was the majority of the 
successful candidate—or, how many votes did he receive 
more than the other? Ans. 2824 votes. 

7, Washington died in 1799, at We age of 67. In what 
year was he born? 

8. In a state containing 2311786 inhabitants, there are 
1143683 females; how many males are there? 
Ans. 1168103 males. 


36 SUBTRACTION. 


9. A lady buys a house for $3000. She spends $169 on 
it for repairs, and then sells it for $3450. Does she gain or 
lose, and how much? Ans. Gains $281. 

10. In 1854, 8088955 bushels of wheat were received in 
Chicago. In 1857, the receipts were 10554761 bushels. What 
was the increase in the 3 years? Ans. 7515806 bushels. 

11. A house and lot are sold for $10575. If the lot cost 
$3250, and the house $8195, does the owner gain or lose, and 
how much? Ans. Loses $870. 

12. The population of the city of New York in 1860 was 
814287, which was 184477 more than it was in 1855. What 
was the population in 1855? Ans. 629810. 

18. Two persons are 375 miles apart. They travel to- 
wards each other, one going 93 miles and the other 57. How 
far are they then apart? Ans. 225 miles, 

14, A man owns a house valued at $6250, and stock to 
the amount of $4500. If he is in debt $3999, what is he 
worth in all? Ans. $6751. 

15. If a merchant who has 19000 bushels of corn, sells 
one customer 4550 bushels, and another 398, how many 
bushels has he left? Ans. 14052 bushels. 

16. A sells B 60 bushels of wheat for $74, a horse for 
$150, a wagon for $95, and $87 worth of butter. B pays 
$125 cash. How much does he still owe A? Ans. $231. 

17. A farmer lays up 237 pounds of butter from his 
own dairy, and buys 349 pounds more of a neighbor. After 
giving away 50 pounds and selling 488, how much has he 
left ? - Ans. 48 pounds. 

18. In a library of 1594 volumes, there are 727 French 
books, 586 German, and 138 Spanish. The rest are Italian; 
how many of the latter are there? Ans. 148 volumes. 

19. From ten thousand subtract seventy-five. Ans. 9925. 


can 


MULTIPLICATION. 37 


MULTIPLICATION. 
63. What will 4 pies cost, at 6 cents each? 


If 1 pie costs 6 cents, 4 pies will cost 4 times 6 cents, or 
24 cents. Here we are required to take 6 four times. This 
process is called Multiplication. 

64. Multiplication is the process of taking a num- 
ber a certain number of times. 


MottiPLicATION TABLE. 


0 taken any number of times is 0. Once 0 is 0; twice 0 is 0; &c. 
© times any number is 0. 0 times 1 is 0; 0 times 2 is 0; &c. 
Once any number is that number. Once 1 is 1; once 2 is 2; &c. 


Twice 8 times | 4 times | 5 times | 6 times | 7 times 
Lime Ooh teig Sixt is 4b lis. 5) Lisc 6) Pis.% 
mage @ is 6.| 2. is 28 | 2 18.10\|/':2.is 12 4}. 2 is 14 
meee. os is 9 | 8.is 12 1° 3 is 1b; 8 iss } 8 is, 21 
4is 8 | 4 is 12 4 is 16 4 is 20 418s 24 | 4 13°28 
5 is 10 5 is 15 5 is 20 5 is 25 5 is 30 5 is 85 
6 is 12/} 6is 18] 6 is 24] 6is 380] 6 is 86| 6 is 42 
716-14)" % is 21°) 7 is 28 | 7 is 85 | 7 is 42 | 7 is 49 
8 is 16 | 8 is 24 | 8 is 82) 8 is 40] 8 is 48 | 8 is 56 
9is 18 | 9 is 27 | 9 is 86| 9is 45] 9 is 54] 9 is 68 
10 is 20 | 10 is 30 | 10 is 40 | 10 is 50 | 10 is 60 | 10 is 70 
11 is 22 | 11 is 83 | 11 is 44 | 11 is 55 | 11 is 66 | 11 is 77 
12 is 24 | 12 is 86 | 12 is 48 12 is 60 | 12 is 72 | 12 is 84 
8 times 9 times 10 times 11 times 12 times 
VR jighs 38 Pigs 69 1 is 10 1 is 11 tis 32 
2 is 16 2 is 18 2 is 20 2 is wiQ2 2. isk 24 
ays, 24 3 is 27 3 is 80 3 is 33 3 is 86 
4 is 32 4 is 86 4 1s 40 4 is 44 4 is 48 
5 is 40 5 is 45 5 is 50 5 is 55 5 is 60 
6 is 48 6 is 54 6 is 60 6 is 66 Gr ts 272 
Peete sis . 6S |). 7 is 70 - 7 is 7 | 7 is | 84 
8 is 64 8 is 72 8 is 80 8 is 88 8 is 96 
viris, 79 9 is 81 9 is 90 9 is 99 9 is 108 


38 MULTIPLICATION. 


65. The number to be multiplied, is called the 
Multiplicand. That by which we are to multiply, 
is called the Multiplier. The result, or number ob- 
tained by multiplication, is called the Product. 

3 times 2is 6. 2 is the multiplicand, 3 the mul- 
tiplier, 6 the product. 


66. The multiplicand and multiplier are patted 
Factors of the product. 2 and 8 are factors of 6. 


67. Multiplication is denoted by a slanting cross’ 


x, placed between the factors. 2x3 is read, and 
denotes, two multiplied by three. 


68. The multiplier shows how many times the 
multiplicand is to be taken. Multiplying 2 by 3 is 
taking 2 three times: 2+2+2=6. 2x3=6. 

Multiplying is therefore a short way of adding a 
number to itself. 


How many trees are there in 

three rows of four trees each? S¢ < & 
We may do this sum in two 

ways. We may say, If one row a ae 

contains 4 trees, 8 rows will con- 

tain 3 times 4 trees, or 12: trees. 

This is doing it by multiplication. 
Or we may say, As there are 4 trees in each row, in the 

three rows there are 4+4+44 trees, or 12 trees. This is 

doing it by addition. The result is the same. 


63. Repeat the example given. "What are we here required to do? 
“What is this process called ?—64, What is Multiplication? How much is 0, 
taken any number of times? How much is 0 times any number? How 
much is once any number? Recite the Table.—65. What is the number to 
be multiplied called? ‘What is that by which we = to multiply called? 
What is the result obtained by multiplication called ?—66. What are the 
multiplicand and multiplier called ?—67. How is multiplication denoted ?— 
68. What does the multiplier show? Multiplying is a short way of doing 
what? Illustrate this. 


i re 


MENTAL EXERCISES. 39 


69. When two numbers are to be multiplied to- 
gether, it makes no difference in the result which is 
taken as the multiplicand, and which as the multi- 
per.” 4x3 12: 3x4=12. 

We have 12 trees in the above engraving, whether we 
take them crosswise as forming 3 rows of 4 each, or length- 
wise as forming 4 rows of 3 each. 

70. The product is of the same kind as the mul- 
tiplicand. 38 times 2 men is 6 men; 8 times 2 
apples is 6 apples; &c. 


MENTAL EXEROISES. 

1. If there are seven days in a week, how many days are 
there in nine weeks? 

Mopet.—If there are 7 days in 1 week, in 9 weeks there 
are 9 times 7 days, or 63 days. Answer, 63 days. 

2. If a man’s wages are three dollars a day, how much 
will he earn in five days? How much in ten days? 

3. If a stage makes 6 trips a day, of 3 miles each, how 
many trips will it make in a week, excluding Sunday ? 

4, What is the cost of 11 tons of coal, at $6 a ton? 

5. How long will it take one boy to do a job which it 
takes five boys eight days to do? 

6. At the rate of 4 for a cent, how many crackers can be 
bought for 7 cents? How many for 12 cents? , 

7. Four quarts make a gallon. How many quarts will a 
five-gallon jug hold? : 

8. What will 12 tables cost, at $11 apiece? 

9. If an omnipus carries 12 passengers each trip, how 
many does it carry in 6 trips? In 8 trips? 


69. In multiplying two numbers, what is found to make no difference? 
Illustrate this with the engraving.—70. Of what kind is the product? 


40 MULTIPLICATION. 


10. If a box of tea lasts 8 persons 11 weeks, how long 
will it last 1 person at the same rate? 

11. There are 2 pints in a quart. How many pints in 3 
quarts? In 5 quarts? In 7 quarts? In 9 quarts? 

12. If two boys do 4 sums apiece every day, how many 
will both do in 8 days? 

18. Three fields contain 3 trees each. Under each tree 
8 cows are lying. How many cows are in the three fields? 

14. If 2 apples can be bought for 1 cent, how many can 
be bought for 10 cents? 

15. Four girls have 2 hens each, and each hen has 6 
chickens. How many chickens have they altogether? 


EXAMPLES FOR THE SLATE. 


71. Multiply 60123 by 8. 

Here the multiplier is a single figure. Set it 
under the units’ figure of the multiplicand. 
Beginning at the right, multiply each fig- 60123 


ure of the multiplicand by the multiplier, 3 
setting each product in a column with the 180369 
figure multiplied. 


Three times 3 is 9; 3 times 2 is 6; 3 times 1 is 3; 8 times 
0is0; 3 times6is18. The last product consists of two fig- 
ures. Set it down with its right-hand figure under the figure 
multiplied. Answer, 180369. 


(1) (2) (3) (4) 
Multiply 9432 71232 52122 81010 
By 2 3 4. 5 


5. At 7 cents a pound, what will 1010 pounds of cheese 


cost? ; 
6. If a man lays up $510 a year, how many dollars will 


he be worth in 9 years? Ans. $4590. 


CARRYING. Al 


~ Carrying. 


72, When a figure is multiplied, the product 
may consist of two figures. As we can not set 
them both down under the figure multiplied, we 
place the right-hand figure there, and add the other 
figure to the next product. 

This adding -of the left-hand figure to the next 
product is called Carrying. a 

ExampLE.—Multiply 7608 by 7. "608 

Begin at the right. 7 times 8 is 56,—5 tens y 
and 6 units. Set the 6 units in the units’ place, 53956 
and carry the 5 tens to the next product. ‘ 

7 times 0 tens are 0 tens, and 5 makes 5 tens. Set it down. 

7 times 6 hundreds are 42 hundreds,—or, 4 thousands and 
2 hundreds. Set the 2 hundreds in the hundreds’ place, and 
carry the 4 thousands to the next product. 

. 7 times 7 thousands are 49 thousands, and 4 makes 53. 
This being the last product, set down both figures. Answer, 
53256. 

73, Rutz ror Carryine.—When any product ex- 
ceeds 9, set down the right-hand figure wm the same 
column with the figure multiplied, and add the re- 
maining figure or figures to the next product. 


EXAMPLES FOR THE SLATE. 


1. 23618 x 6. Ans. 141708. 5. 289467385 x 9. 

2. 41726 x 7. Ans. 292082. 6. 9156738 x8. 

8. 683918547 x 4. 7. 4507956 x6. 

4, 9254658973 x5. 8. 51708949326 x 8. 

72. When, on multiplying a figure, we get a product of two figures, what 


must we do? What is this process called? Illustrate the mode of carrying, 
with the example given.—73. Recite the rule for carrying in multiplication. 


42 MULTIPLICATION. 


9. The multiplicand is ninety-three thousand and forty- 
six; the multiplier is 83; what is the product? Ans. 279188. 
10. How much is nine times seven hundred and fifty-eight 
thousand and twenty-nine? Ans, 6822261. 
11, There are 865 days in a year. How many days are 
there in three years? ; 
12. The moon is 240000 miles from the earth. If there 
were a comet 6 times that distance from the earth, how far 
would it be? Ans. 1440000 miles. 
18. Nine million eight hundred thousand two hundred 
and fifty-seven is one factor; seven is the other; what is the 
product? Ans. 68601799. 


Multiplying by two or more figures, 


74. Multiply by 10, 11, and 12, in one line. 


Exampie.—Multiply 541 by 12. BAL 

12 times 1 is 12; set down 2 andcarry 1. 12 19 
times 4 is 48, and 1 makes 49; set down 9 andcarry ——~— 
4, 12 times 5 is 60, and 4 makes 64. Set it down. 6492 
Answer, 6492. 


75. When the multiplier is over 12, amt by 
its figures separately. 

Exampity.—Multiply 287 by 156. 

We can not multiply by 156 at once. Hence we 
first multiply by the 6 units; then by the 5 tens, 
or 50; then. by the 1 hundred. Thus we get three 
Partial Products, as they are called; and, eed 
these, we get the whole product. 

74. How are we to multiply by 10, 11, and 12? Give an example.—75. 


When the multiplier is over 12, how are we to proceed? What are the re- 
sults obtained by multiplying by each figure called ? 


RULE.—PROOF. 43 


Multiplicand 287 
Multiplier 156 
ere. 1722 =287x 6 
1435  =287x 50 
Products 


Q8T == 287x100 
Product 44772 =087x156 


76. General Rule. 


1. Set the multiplier under the multiplicand, 
units under units, tens under tens, ke. 

2. Beginning at the right, multiply by each figure 
of the multiplier im turn, setting the first figure of 
each partial product in a column with the figure 
used in multiplying. 

3. Add the partial products, and their sum will 
be the whole product. | 


Proof. 


77. Multiply the multiplier by the multiplicand. 
If this product is the same as the former 156 
one, the work is right. 287 

ExamPpLe.—Prove the example given above. 1092 

Multiply the multiplier 156, by the multiplicand 1248 » 
287. The product is 44772, which is the sameas 342 
the former product. Hence the work is right. AAT9 


78. When two numbers are to be multiplied to- 
gether, it is usual to take the one with the fewer 
figures for the multiplier. 

Multiply 287 by 156.—76. Recite the general rule for multiplication.— 


77. How is multiplication proved? Prove the example last given.—78. Which 
of two numbers is it usual to take for the multiplier? 


44 _ MULTIPLICATION. 


EXAMPLES FOR THE SLATE. 
Find the value of the following. Prove each example. 


1. 356x11. 5. 1678 x 948. 
2, 289 xy, 6. 3548 x 751. 
8. 165 x 234. 7, 2674 x 863. 
4, 829x576. 8. 9463 x 594. 

9. How much is 42 times 6257 ? Ans. 262794. 
10. How much is 53 times 8164? Ans. 432692. 
11. Multiply 4567031 by 147. Ans. 6713853557. 
12. Multiply 4905604 by 263. Ans. 1290178852. 
13. Multiply 65728918 by 5674. Ans. 372945852362. 
14. Multiply 8549412 by 32895. Ans, 281232907740. 
15. Multiply 5076398 by 9168. Ans. 46540416864. 
16. Multiply 765987 by 7896. Ans. 6048233352. 
17. Multiply 542896 by 6892. Ans. 3741639232. 
18. Multiply 99887 by 2678. Ans. 266997951. 


19. Find the product of six thousand four hundred and 
twelve, and seventy-five thousand eight hundred and thirty- 


nine. Ans. 486279668. 
20. Multiply eighty thousand four hundred and sixty- 
seven, by nine hundred and seventy-six. Ans. 78535792. 


21. In a certain orchard, there are 15 rows of trees, and 
19 trees in each row. How many apples will the whole 
orchard produce, if the average yield of each tree is 948 
apples? Ans. 270180 apples. 


® im the multiplier. 


79. When 0 occurs in the multiplier, bring it 
down, and go on multiplying by the next figure, all 
en the same line. 


79. When 0 occurs in the multiplier, how are we to proceed ? 


NAUGHT IN THE MULTIPLIER. 45 


ExampLe.—Multiply 2473 by 5008. 


Beginning at the right, multiply by 8. Then iat 
bring down the first 0 of the multiplier in the ——_ 
tens’ place, and the second 0 in the hundreds’ 19784 


place. Then multiply by 5, placing the product 1236500 
in the same line. Finally, add the partial prod- 12884784 
ucts. Ans., 12384784. 


EXAMPLES FOR THE SLATE. 


1. Multiply 46893 by 40308. Ans. 1890163044. 
2. Multiply 962734 by 700906. Ans. 674786037004. 
3. Multiply eighty thousand and twenty-nine by five thou- 
sand and seven. Ans. 400705208. 


4, The Morris and Essex Canal, which is 101 miles in 
' length, cost on an average $30693 a mile. What was the 
whole cost? Ans. $8099993. 

| 5. If 19008 pounds of hay are required for the horses of a 
cavalry regiment one day, how many pounds will be n®eded 
for 206 days? 

6. What would be the cost of constructing 309 miles of 
plank road, at $3975 a mile? 

7. How many apples will an orchard containing 208 trees 
produce, if the average yield is 1269 apples for each tree? 

8. In 8 editions of 750 books each, how many pages, if 
each book contains 407 pages? Ans. 915750 pages. 


First find how many books there are, then how many pages. 


Naughts at the right. 


80. When there are naughts at the right of erther 
factor or both, multiply the other figures, and annex 
to their product as many naughts as are at the right 
of both factors. 


Solve the example given above.—80. When there are naughts at the right 
of either factor or both, how are we to proceed ? 


46 MULTIPLICATION. 


Cutting off the naughts, multiply 45 by 37\0 
37. Then, as there are four naughts at the 315 
right of the multiplicand and one at the 135 
right of the multiplier, annex five naughts §= ————— 
to the product. Answer, 166500000. 166500000 


81. Multiplying a number by 1 does not affect its 
value. Hence, to multiply by 10, 100, 1000, c&c., 
sumply annex as many naughis as are in the mul- 
tuplier. 

87x10=870  87x100=8700  37x1000=87000 
EXAMPLES FOR THE SLATE. 


Find the value of the following :— 


1. 6706 x10. 6. 8000 x 700. 
» 2. 89271 x 100. 7, 290000 x 98. 
8. 50860 x 120. 8. 568 x 11000. 
4. 7800 x 4300. 9. 74600000 x 56.. 
5. 867 x 10000. 10. 6060 x 7040. 
11. If there are 100 cents in one dollar, how many cents 
are there in $60400 ? Ans. 6040000 cents. 


12. How many men were there in 11 Roman legions, if 
there were 4500 men in one legion? 

18. 100 pounds make a hundred-weight. How many 
pounds in a ton, which contains 20 hundred-weight? 

14, The Roman soldiers on a march carried 60 pounds’ 
weight apiece. How many pounds did 3000 soldiers carry ? 

15. What will five hundred acres of land cost, at thirty 
P dollars an acre? > ~. Ans. $15000. 
16. What will 1000 acres cost, at $20 an aere? 


: om 
Multiply 450000 by 370.—81. What is the effect of multiplying a number 
by 1 To multiply by 10, 100, 1000, &c., what are we to do? 


MULTIPLYING BY FACTORS. 47 


Multiplying by factors. 


82, When the multiplier is itself a product, we 
may multiply either by the whole, or by its factors 
in turn. The result will be the same. 

Exampte.—Multiply 84 by 36. 


36=6 x6 or, 9x4 or, 12x38. 


84. 84 84. 84. 
36 oe 9 12 
504, 504. 156 1008 
252 6 4 3 
3024. 3024. 3024. 3024. 


Whether we multiply by 36 at once or by the different 
sets of factors that produce it, we get the same product,— 
3024. Multiplying by factors, therefore, proves whether the 
product first obtained is right. 


EXAMPLES FOR THE SLATE. 


In these examples, first multiply by the multiplier at once. 
Then prove the result by multiplying by its factors. 


1. 875x81 (9x9). 4, 9000 x 27. 
2. 5761x44 (11x4). B. 18274. 24. 
8. 486 x72 (6x12 or 8x9). 6. 45000 x 20. 


7. How many ounces in 258 pounds of sugar, if there are 
. 16 ounces in one pound? 


8. How many bricks are there in 18 loads, if each load 
contains 1250 bricks? 
9. What will 28 horses cost, at $75 apiece? 
10. How many pounds in 82 firkins of butter, allowing 
56 pounds to the firkin? 


82. When the multiplier is itself a product, what two modes of proceed- 
ing aro there? Illustrate these, with the given example. 


48 MULTIPLICATION. 


MISOELLANEOUS EXAMPLES. 


1. The sun is 95000000 miles from the earth; the moon 
is 240000 miles. How much farther from us is the sun than 
the moon? Ans. 94760000 miles. 

9. A man worth $10000 gains $1000 by one sale, and loses — 
$500 by another.. What is he then worth ? Ans. $10500. 

3. If the railroad from Albany to Buffalo, 326 miles in 
length, cost $25649 a mile, what was the whole cost? 

Ans. $8861574. 

4. How far will a locomotive travel in 7 days of 24 hours 
each, if it goes 30 miles an hour? Ans. 5040 miles. 

5. A man left each of his 3 daughters 510 acres of land, 
and his wife 100 acres more than all his daughters together. 
What was the wife’s portion? Ans. 1680 acres. 

6. A farmer raised 1570 bushels of potatoes, and bought 
730 bushels more. After selling 4 lots of 500 bushels each, 
how many bushels had he on hand? _ Ans. 300 bushels. 

7. How many hours in 865 days of 24 hours? Ans. 8760. 

8. The earth moves in its orbit 68000 miles an hour; how 
far will it move in 365 days? Ans. 595680000 miles. 

9. The President’s salary is $25000. If he spends $19500 
a year, how much will he save during his 4 years’ term? 

How much will he save inl year? How much, then, in 4 years? 

; Ans. $22000. 

10. A has 1060 sheep; B has 849; O has 1276. How 


many must C buy, fo have as many as A and B together ? 
Find how many A and B have. Then find the difference between this 
number and C’s, 
Ans. 633 sheep. 
11. Daniel Webster was born in 1782 and ¥died in 1852.” 
es 
How old was he? - 
12. At $3 a yard, what’ will 101 pieces of cloth cost, if 
each piece contains 40 yards? — Ans. $12120, 


MISCELLANEOUS EXAMPLES. 49 


13. If a ship sails east from port 850 miles a day for 4 
days, and is then driven west 289 miles, how far is she from 
port ? . Ans. 1111 miles. 

14. John has 31 marbles; James has 6 times as many; 
Henry has as many as John and James together. How many 
marbles have they all? Ans. 484 marbles. 

15. What cost 100 wagons, at $75 apiece? 

16. Find the product of 8089 and 9007. Subtract this 
product increased by 377, from 100000000. Ans. 271420000. 

17. Trinity Church, N. Y., is 284 feet high. St. Peter’s at 
Rome is 166 feet higher; what is its height? 

18. A man who has $1201 in the bank, draws out enough 
to pay for 5 lots of land at $195 a lot. How much is left in 
the bank? Ans. $226, 

19. North America contains 8000000 square miles; South 
America, 7000000; Asia, 1000000 more than both Americas © 
together. How many square miles in Asia? Ans. 16000000. 

20. An army of 12100 men lost 631 in killed and wounded, 
and twice that number taken prisoners. Mow many were 

» > left? . Ans. 10207 men. 
“21. In a school of 127 boys, four times as many study 
Arithmetic as study Latin. The Latin class numbers 28; 
how many study Arithmetic? Ans. 112 boys. 

22. What is the cost of a carriage and span of horses, if 

» each horse cost $235, and the carriage $679? Ans. $1149. 

23. A boat makes 208 trips in a season, and carries on an 
average 106 passengers each trip ; how many does she carry 
in all? Ans. 22048 passengers. 

24, Find the cost of 53 mules, at $100 apiece. 

ye o5. A grocer who has 15 firkins of butter, containing 56 
pounds each, sells four of his customers 240 pounds. How 
much has he left? Ans, 600 pounds. 

3 


50 DIVISION. 


DIVISION. 


83. If 2 apples can be bought for 1 cent, how 
many cents will 6 apples cost ? 

2 apples cost 1 cent; hence 6 apples will cost as 
many cents as 2 is contained times in 6. Here we 
are required to find how many times 2 is contained 
in 6. This process is called Division. 


84, Division is the process of finding how many 
times one number is contained in another. 


Drvision TABLE. 


Any number is contained in 0, 0 times. 2 
»| 1 in 1, once. 2 in 2, once.. 3 in 38, once. 

1 in 2, twice. 2in 4, twice. 3 in 6, twice. 
1 in 8, 38 times. 2in 6, 8 times. 3 in 9, 8 times, 
1 in 4, 4 times. 2in 8, 4 times. 3 in 12, 4 times. 
1 in 5, 5 times. 2 in 10, 5 times. 3 in 15, 5 times. 
1 in 6, 6 times. 2 in 12, 6 times. 3 in 18, 6 times. 
1in 7, 7 times. | 2 in 14, 7 times. | 3 in 21, 7 times. 
1 in 8, 8 times. 2 in 16, 8 times. 3 in 24, 8 times. 
1 in 9, 9 times. 2 in 18, 9 times. 3 in 27, 9 times. 


4 in 4, once. 5 in 5, once. 6 in 6, once. 
4 in 8, twice. 5 in 10, twice. 6 in 12, twice. 
4 in 12, 3 times. 5 in 15, 8 times. 6 in 18, 3 times. 
: 4 in 16, 4 times. | 5 in 20, 4 times. | 6 in 24, 4 times. 
4 in 20, 5 times. 5 in 25, 5 times. 6 in 80, 5 times, 
4 in 24, 6 times. 5 in 380, 6 times. 6 in 36, 6 times, 
4 in 28, 7 times. 5 in 35, 7 times. 6 in 42, 7 times. - 
4 in 32, 8 times. |- 5 in 40, 8 times. 6 in 48, 8 times. 
» 4 in 36, 9 times. 5 in 45, 9 times. 6 in 54, 9 times. 


83. Repeat the example. What are we here required to do? What is 
this process called?—84. What is Division? How many times is any num- 
_ ber contained in 0? Recite the Table. 


DIVISION. 51 


Tin Jé) onee, 8 in 8, once. 9 in 9, once. 

7 in 14, twice. 8 in 16, twice. 9 in 18, twice. 

7 in 21, 8 times. 8 in 24, 3 times. | 9 in 27, 3 times, 
7 in 28, 4 times. 8 in 32, 4 times. 9 in 36, 4 times. 
7 in 35, 5 times. 8 in 40, 5 times. 9 in 45, 5 times. 
7 in 42, 6 times. | 8 in 48, 6 times. 9 in 54, 6 times. 
7 in 49, 7 times. 8 in 56, 7 times. 9 in 68, 7 times. 
7 in 56, 8 times. 8 in 64, 8 times. 9 in 72, 8 times. 
7 in 63, 9 times. 8 in 72, 9 times. 9 in 81, 9 times. 
10 in 10, once. 11 in 11, once. 12 in 12, once. 

10 in 20, twice. 11 in 22, twice. 12 in 24, twice. 


10 in 80, 3 times.| 11 in 83, 3 times. | 12 in 386, 38 times. 
10 in 40, 4 times. | 11 in 44, 4 times. | 12 in 48, 4 times. 
10 in 50, 5 times.| 11 in 55, 5 times. | 12 in 60, 5 times. 
10 in 60, 6 times.| 11 in 66, 6 times. | 12 in 172, 6 times. 
10 in 70, 7 times. | 11 in 77, 7 times. | 12 in 84, 7 times. 
10 in 80, 8 times.} 11 in 88, 8 times. | 12 in 96, 8 times. 
10 in 90, 9 times. | 11 in 99, 9 times. | 12 in 108, 9 times. 


85. The number to be divided, is called the Divi- 
dend; that by which we are to divide, the Divisor. 

The result, or number obtained by dividing, is 
called the Quotient. It shows how many times the 
divisor is contained in the dividend. 

2 is contained in 6, 3 times; 2 is the divisor, 6 the divi- 
dend, 3 the quotient. 

86. The divisor is not always contained an exact 
number of times in the dividend. It may go a cer- 
tain number of times, and some over. What is left 
over is called the Remainder. 

5 is contained in 15 exactly 3 times. In 16 it goes 8 times, 
»and 1 over; 3 is the quotient, 1 the remainder. 


85. What is the number to be divided called? What is that by which 
we are to divide called? What is the result obtained by dividing called? 
What does the quotient show? Illustrate these definitions—86. What is 
meant by the Remainder? Give an example. 


52, DIVISION. 


87. Division is denoted by a short horizontal line 
between two dots +. This sign indicates that the 
number before it is to be divided by the one after 
it. 6+2 is read, and denotes, sew divided by two. 


2 


MENTAL EXEROISES. 


How many times is 8 contained in 9? 10in 40? 7in 562 
11in 99? 4in32? 9in81? 6in42? 12in36? 2in18? 
11in 55? 5in80? 12in96? 8in 72? 7inO? 


How many times is 4 contained in 23? (Ans. 5 times, and 
8 over.) 2in 5? 6in27?.9in20? 8in 389? 8in10? 4 
in 29? 5in 38? 10in 44? 6in59? 12 in 75? 


What is the quotient, and what the remainder, in the fol- 
lowing? 34+4. 31+6. 22+8. 108+12. 87+5. 18+8. 
65-9. O+4. 89-+10. 8+5. 48+8. 72+9. 47+11. 72 
+8. 24+3. 24+8. 64212. 28+10. 81+11. 


1. If 64 cherries are divided into 8 equal piles, how many 
will there be in each pile? 

Mopret.—As many cherries as 8 is contained times in 64, 
or 8. Answer, 8 cherries. 

2. Twenty-one cents are distributed equally among 7 beg- 
gars. What is the share of each ? 

8. How many coats, at $5 apiece, can I buy for $20? 

4, John’s father gave him 18 pears, to be divided equally 
among his two sisters and himself. How many did each get? 

5. How many bushels are there in 16 pecks, there being 
4 pecks in 1 bushel? 

6. Twelve make a dozen. How many dozen in 24? 

7. Allowing 8 yards of calico for a dress, how many 
dresses can be made out of 40 yards? 


87. How is division denoted? What does this sign indicate ? 


MENTAL EXERCISES. . 58 


_ 8. If a dozen pineapples are sold for 96 cents, how much 

is that apiece ? 

9. How many stage-coaches, carrying 9 persons each, will 
be needed to carry 45 passengers? 

10. How many ten-ounce bottles will it take to hold 40 
ounces of alcohol ? 

11. There are 49 days in 7 weeks. How many days are 
there in 1 week? 

12. If a family use 42 quarts of milk in a week, how much 
do they use in a day? 

13. If 11 sheep yield 83 pounds of wool, what is the aver- 
age yield for each sheep ? 

14, How many caps, at $2 each, can you buy for $12? 

15. If I go 50 miles in 10 hours, what is my rate per hour? 

16. How many pair will 16 chickens make? 

17. If 48 trees are planted in 6 equal rows, how many 
trees will there be in a row? ; 

18. Eighty-four eggs make how many dozen? 

19. Twelve yards come in a piece of ribbon. How many 
pieces will 60 yards of ribbon make? 


EXAMPLES FOR THE SLATE. 


88. Divide 27036 by 3. 

Here the divisor is'a single figure. Set it at the 
left of the dividend, with a curved line 3) 27036 
between. a ae 

In dividing, always begin at the left. See how 
many times the divisor is contained in each figure 


of the dividend, and set the quotient under the 
figure divided. 


88. When the divisor is a single figure, where is it set? Where must we 
always begin in dividing? How do we find the quotient? 


54 DIVISION. 


8 is not-contained in 2. See, then, how 3) 27036 
often it will go into 27, the first two figures. 9012 
8 in 27,9 times. Set down 9 under 7, the right- 
hand figure of the two divided. 

8 in 0,0 times. Set it down, and remember that 0 must 
never be omitted in the quotient, unless it is the first figure. 

3 in 8, once; set down 1. 8 in 6, twice; set down 2. 
Answer, 9012. 


89, In the following examples, divide as above:— 


() tre (3) (4) 
2) 148264 3) 219063 4.) 28084 5) 40500 


5. Divide thirty-six hundred and six, by six. 
6. Divide fifty-six million by 7: by 8. 


Carrying. 

90. When, after having divided all the figures 
of the dividend, there is a remainder, set it down. 
as such. But if before this a remainder occurs, 
prefix it (nm the mind) to the next figure of the 
dividend, and continue the division. 

This prefixing is called Carrying. 

Examprte.—Divide 265231 by 6. 

6) 265931 
44205 and 1 remainder. 

6 is not contained in 2. 6 in 26, 4 times and 2 over; set 
down 4 under the 6, and carry 2. 6 in 25, 4 times and 1 over; 
set down 4 and carry 1. 6 in 12,;twice; set down 2. 6 in 3, 
0 times and 3 over; set down 0 and carry 3. 6 in 31, 5 times 


and 1 over; set down 5 in the quotient, and 1 as remainder. 
Answer, 44205 and 1 remainder. 


Go through these steps in the given example.—90. What is meant by 
Carrying in division? When is the remainder set down? Show how we 
carry, in the given example, 


CARRYING.—PROOF. 55 


91. Kure ror Carryine.—Jf, in dividing any 
Jigure of the dividend except the last, a remainder 
occurs, prefix vt (in the mind) to the next figure to be 
divided, and continue the division. 


92, When the divisor is greater than the figure 


to be divided, set 0 in the quotient, and carry the 
latter figure. 


Proof of Division, 
93. Multiply the quotient by the divisor, and 
add in the remainder, if there is one. If the result 
equals the dividend, the work is right. 


Exampire.—Prove the example given on the 44905 
last page. 6 
Multiply the quotient 44205, by the divisor 965930 ' 
6. Add in the remainder, 1. The result is ae, 
265231, which is the same as the dividend. 
Hence the work is right. 265231 


EXAMPLES FOR THE SLATE. 
Find the quotient and remainder. Prove each example. 


1. 50786+2. Ans. 25368. 6. 117860194-+-9. Rem. 5. 
2. 271216+-8. Ans. 33902. 7. 2190076387+4. Rem. 1. 
8. 124555806-+8. 8. 470169628+8. Rem. 4. 
4, 2501024085 +5. 9. 401018116+6. Rem. 2. 
5. 5068595742+7. 10. 3089138211+5. Rem. 1. 
11. Divide one million four hundred thousand six hundred 
and twelve, by 9. Ans. 1556238, 5 rem. 


12. The dividend is fifteen million; the divisor is seven; 
find the quotient. Rem. 1. 
91. Give the rule for carrying in division.—92. What must be done when 


the divisor is greater than the figure to be divided ?—93. How is division 
proved? Prove the example just given. 


56 DIVISION. 


13. How often is 3 times 2 contained in one hundred and 
forty million and four? Ans. 233333834 times. 

14. If it costs $17068 to build four miles of plank road, 
what will one mile cost? 

15. A man leaves $317520 to be divided equally between 
his two children. What is the share of each? 

16. John gathers 761 nuts, and Jacob 848. If they share 
them equally, how many will each have? Ans. 802 nuts. 

17. How many six-acre fields can be laid out in a planta- 
tion of 1488 acres? 


94, The mode of dividing shown above is called 
Short Division. In Short Division, the carrying is 
done in the mind, and the quotient is written under 
the dividend. | 


Dividing by two or more figures, 
95. Divide by 10, 11, and 12, by short division. 


ExaMPLe.—Divide 1084608 by 12. 

12 is not contained in 1, orin 10. Hence 
we take the first three figures. 12 in 108, 12) 1084608 
9 times; set down 9 under 8, the right-hand 90384 
figure of those divided. 

12 in 4,0 times; set down 0. 12 in 46, 3 times and 10 
over; set down 8 and carry 10. 12 in 100, 8 times and 4 
over; set down 8 and carry 4. 12 in 48, 4 times; set down 
4, Answer, 90384. 


Find the value of the following :— 

1. 8314607+10. Rem. 7. | 4. 1109964+12. Ans. 92497. 
2. 9923166+11. Rem. 0. | 5. 1018193+11. Ans. 92563. 
3. 1078285+10. Rem. 5. | 6. 1198872+12. Ans. 99906. 


94, What is this mode of dividing called? In Short Division, how is the 
carrying done? Where is the quotient written?—95. How are we to divide 
by 10, 11,12% Give an example. 


LONG DIVISION. 5G 


96. When the divisor is over 12, proceed by what 
is called Long Division. 


97. In Short Division, we set the quotient wader 
the dividend. In Long Division, we place it at the 
right, with a curved line between. 

In Short Division, we find what is to be carried, 
and prefix it to the next figure, 7m the mind. In 
Long Division, we multiply, subtract, and prefix as 
before; but we write down all the figures used. 

ExamMpiLe.—Divide 9594 by 39. 

: The quotient is now to be set at the 
Bd: 9594 (246 right of ihe dividend. Beginning at the 


sx2— (8 left of the dividend, take as many figures 
179 as are required to contain the divisor—in 
39x4=156 this case, two. We find on trial that 39 
934. is contained in 95 twice. Set 2 in the 
a9 xe 934 quotient as the first figure. 


Multiply the divisor by this 2. Twice 
39 is 78; set the product under 95, and subtract. The re- 
mainder is 17, which (as in short division) we prefix, by 
bringing down 9, the next figure of the dividend. 

- Now repeat the same steps. Find how often 39 is con- 
tained in 179. It goes4times. Set 4in the quotient; mul- 
. tiply the divisor by it; set the product under 179, and sub- 
tract. The remainder is 23, to which bring down 4, the next 
figure of the dividend. 

Find how often 39 is contained in 234. It goes 6 times. 
Set 6 in the quotient; multiply the divisor by it; set the- 
product under 234, and subtract. There is no remainder. 
As we have now brought down all the figures of the dividend, 
the sum is finished. Answer, 246. 

95, 179, and 234, axe called Partial Dividends. 


96. What process do we use when the divisor is over 12?—97. Show how 
Long Division differs from Short Division. Divide 9594 by 89, explaining the 
several steps. 


8% 


58 DIVISION. 


98. We may not always, on the first trial, get 
the right quotient figure. 

If, when we multiply the divisor by any quotient 
figure, the product comes greater than the partial 
dividend, the quotient fpr is too great, and must 
be Nicaea: 

If, on the other hand, we have a remainder 
greater than the divisor, the quotient figure is too 
small, and must be increased. 


39) 9594(8 Exampes.—In the last example, if we 

117 say 39 is contained 8 times in 95, we get a 

product greater than the partial dividend, 

and can not subtract. We must therefore diminish the quo- 
tient figure. 

If we say it is contained once, on multi- 39) 9594 (1 
plying and subtracting we get a remainder 39 
greater than the divisor. We must therefore 56 
increase the quotient figure. 


99. For every figure of the dividend brought 
down, a figure must be placed in the quotient. To 
prevent mistakes, it is well to place a dot under 
each figure as it is brought down. 

100. When the divisor is not contained in the 
partial dividend, set 0 in the quotient, and bring 
down the next figure of the dividend. 

If several figures are brought down before the 
divisor will go into the partial dividend, set a 0.4 in 
the quotient for each. 


What are the numbers formed each time after a figure is brought down 
called ?—98. In what are we liable to make mistakes? When may we know 
that the quotient figure is too great? When, that it is too small? Give 
examples.—99. To prevent mistakes in bringing down the figures, what is 
recommended ?—100. When the divisor is not contained in the partial divi- 
dend, what must be done? 


EXAMPLES FOR PRACTICE. 59 


Exampie.—Divide 172602 by 86. 
86) 17 9602 (2007 Here our first remainder is 0. 
172 


Bring down 6. 86 in 6, 0 times; set 
Sa 0 in the quotient, and bring down 
602 ‘ : 
609 the next figure. _ 86 in 60, 0 times; 
ets set another 0 in the quotient, and 
bring down 2, the next figure. 86 in 602, 7 times. Set 7 
in the quotient; multiply and subtract. Answer, 2007. 


EXAMPLES FOR THE SLATE. 


Prove each example. In the first nine, if your work is 
right, the remainder will come equal to the quotient. 


1. Divide 6765 by 122. 10. 259828-+84, Ans. 7642. 
2. Divide 80157 by 846. | 11. 70388+57. Ans. 1284. 
8. Divide 34075 by 234. | 12. 589902-+23. Rem. 0. 
4. Divide 159750 by 425. | 18. 470112+354. Ans. 1828. 
5. Divide 269459 by 576. | 14. 16199569+1848. Rem. 1. 
6. Divide 39298 by 801. | 15. 14003794-+-6974. Rem. 2. 
%, Divide 466281 by 926. | 16. 18873585-+2643. Rem. 5. 
8. Divide 854080 by 1087. | 17. 1288025-+1536. Rem. 9. 
9. Divide 3054436 by 2953.| 18. 789783--19263. Rem. 0. 
19. Divide 24280295 by 9684. Rem. 2507. 
20. Divide 1521808704 by 234. Ans. 6503456. 
21. Divide 5763447 by 678509. Rem. 835875. 
22. Divide 24280830966 by 604. Rem. 162. 
93. Divide 48423768038 by 807009. Rem. 2. 
24. Divide 22687013 by 810781. Ans. 78. 
25. Divide 78204820 by 8679548. Rem. 88888. 
26. Divide 36132610 by 8603. Rem. 10. 
27. Divide 1651877824 by 6503456. Ans, 254. 
28. Divide 7807085 by 945. Rem. 440. 
29. Divide 4848708 by 808. Rem. 708. 
30. Divide 1917183 by 9682. Rem. 147. 


60 DIVISION. 


General Rule for Division. 


1. Sct the divisor at the left of the dividend, with 
a line between. 

2. Take as many figures at the left of the dwi- 
dend as will contain the divisor, and jind how 
many times wt will go into them. 

8. If the divisor zs 12 or less, set this first quo- 
tient figure under the figure divided, or under the 
right-hand jigure of those divided, if more than one 
are taken. Divide each figure of the dividend in 
turn, carrying what zs over, and setting each quo- 
tient figure under the figure divided. 

A, If the divisor is over 12, set the first quotient 
figure at the right of the dividend. Multiply the 
divisor by it, and subtract the product from the par- 
cial dividend. } 

5. Bring down the next figure of the dividend. 
Find the next quotient figure, multiply, and sub- 
tract, as before. Go on thus, tull all the figures of 
the dividend are brought down. 

6. Lf any partial dwidend is too small to contam 
the divisor, set 0 in the quotient, bring down the 
next figure, and go on as before. 


EXAMPLES FOR THE SLATE. 


1. The greatest height reached by man in a balloon is 
23027 feet. How many miles is this, allowing 5280 feet to 
the mile? Ans. 4 miles, 1907 feet. 

2. The earth’s circumference is about 25000 miles. How 
many days would it take a person to traverse it, going at the 
rate of 125 miles a day? 


EXAMPLES FOR PRACTICE. 61 


8. How many barrels of apples, at $3 a barrel, can be 
bought for $2568? If one tree produces 8 barrels, how many 
trees will it take to yield this quantity ? Ans. 107 trees. 

4, If a ship bound from New York to Canton sails 9225 
miles, and is 75 days making the voyage, how many miles a 
day does she average? | 

5. The earth’s distance from the sun is about 95000000 
miles. It takes a sunbeam about 8 minutes to reach the 
earth; how many miles does light travel in a minute? 

Ans. 11875000 miles, 
_ 6. A flour barrel holds 196 pounds of flour. How many 
barrels will it take to hold 406700 pounds? Ans. 2075 bbl. 

7. If 47 miles of railroad cost $1815375, what is the cost 
per mile? Ans. $38625. 

8. The product of two factors is 3923828687. One of the 
factors is 46391; what is the other? * Ans. 8457. 

9. Twenty-six times a certain number is 1580540. What 
is the number ? Ans. 60790. 

10. In 1850 there were 2526 newspapers and periodi- 
cals in the United States, of which 426409978 copies were 
printed yearly. What was their average yearly circula- 
tion? Ans. Over 168808 copies. 

11. If 218669 bushels of rye are stored in equal lots in 11 
warehouses, how much is there in each? Ans. 19879 bu. 

12. How long will 15000 pounds of flour last a garrison 
of 250-men, allowing them 750 pounds a day? Ans. 20 days. 

13. How many impressions can a printing press make in 
one hour, if it makes 633240 impressions in one day of 24 
hours? Ans. 26385 impressions. 


* Note.—When a product and one of its factors are given, to find the 
other, divide the product by the given factor. 
3x9= 27 Then 27+3=9 Or, 27--9=8 


62 DIVISION. 


Naughts at the right of the divisor. 


101. When there are naughts at the right of the 
divisor, cut them off, and also as many figures at 
the right of the dividend. 

Divide the remaining figures of the dividend by 
those of the divisor. Lf there is a remainder, annex 
to it the figures cut of from the dividend for the 
true remainder; of not, the figures cut off are the 
true remainder. 


Divisor Dividend Quo. Divisor Dividend Quo. Divisor Dividend 
269 ) 5399 (20 A799) 23716 (5 3009 ) 189991 
52 235 7 Ae 
RTA pies 63 Quo. 

19 2 


Ans. 20, 190 rem. Ans. 5, 216 rem. Ans. 63, 1 rem. 


102. Dividing a number by 1 does not change its 
value. Hence, to diwide a number by 10, 100, 1000, 
&c., simply cut off as many figures at the right of 
the dividend as there are naughts in the divisor. 
The remaining figures are the quotient; those cut 
off, the remainder. 


3500+-10=350  3500+100=35 3500+1000=8, 500 rem. 


EXAMPLES FOR THE SLATE. 


1. Divide 186740 by 10. 6. 2294003--3700. Rem. 8. 
2. Divide 729500 by 100. 7. 854009+14000. Rem. 9. 
8. Divide 729500 by 1000. 8. 1782000+9900. Lem. 0. 
4, Divide 729500 by 10000.| 9. 405349+490. Rem. 119. 
5. Divide 901109 by 1000. |.10. 253579+-510.. Mem. 109. 


101. How must we proceed when there are naughts at the right of the 
divisor ?—102. What is the effect of dividing a number by 1? Give the rule 
for dividing by 10, 100, 1000, &c. 


EXAMPLES FOR PRACTICE. 63 


11. About three million tons of iron are produced yearly 
in Great Britain, which is twenty times as much as is pro- 
duced annually in Sweden. How much is produced in 
Sweden? Ans. 150000 tons. 

12. A ship bound from Boston to Australia, a voyage of 
13000 miles, sails at the rate of 100 milesaday. How long 
is she on the passage ? 

18. The product of two factors is 238700. One of the 
factors is 770; what is the other? Ans. 310. 

14. A cotton crop of 26320 pounds was put up in bales 
averaging 560 pounds each. How many bales did it make? 

15. Sound moves 1120 feet in a second. How long before 
we hear a cannon, fired at a distance of 12320 feet? 


MISCELLANEOUS EXAMPLES FOR THE SLATE. 


To find a sum, add. 

To find a difference, subtract. 
To find a product, multiply. 
To find a quotient, divide. 

1. Find the sum, then the difference, then the product, 
then the quotient, of 125 and 875. 

2. A person who had 200 acres in one state, 1173 in an- 
other, and 127 in a third, divided his land equally among his 
wife and four sons. How much did each receive? 

Find how much land he had in all. Among how many did he divide it? 

3. If a clock ticks 3600 times in 1 hour, how often will it 
tick in two days of 24 hours each ? Ans. 172800 times. 

4, How many times is 20 x 50 contained in 50 x 20? 

5. A barrel of flour contains 196 pounds. If a family of 
5 persons use 4 pounds of flour a day, how long will a barrel 
last them ? : Ans, 49 days. 

6. Three men put in $650 each, and invest the whole in 
salt, at $2 a sack. How many sacks do they buy? Ans. 975. 


64 = EXAMPLES IN ADDITION, SUBTRACTION, 


7. A merchant worth $43000, after losing $1750 in trade, 
invested the rest in land, at $10 an acre. How many acres 
did he buy? Ans. 4125 acres. 

8. What is the difference between 16 x 49 and 49x8x2? 

9. A lady who had eight children, lost two of them. 
Among the survivors she divided her property; which con- 
sisted of $1200 cash, $21550 in stock, and $7250 in bonds. 
What was the share of each? Ans. $5000. 

10. The sum of three numbers is 27846. One of the num- 


bers is 11587; the second is 596; what is the third? 


Find the sum of the two given numbers. The third will be the differ- 
ence between their sum and the whole sum. 
Ans. 15663. 


11. A man leaves his three sons $36000. To the first he 
leaves $14655; to the second, $9875; how much does he 
leave the third ? Ans. $11470. 

12. A soapboiler makes 5977 pounds of soap, of which 
657 pounds are soft soap, while the rest is in bars. He wishes 
to pack his bar soap in boxes holding 70 pounds each. How 


many boxes must he procure? Ans. 76 boxes. 
13. How much must I add to 40 times 60, in order to 
make 3000? . 


How much is 40x60? What is the difference between this and 3000? 
14. A farmer wishes to buy some land for $720. He lays 
up $9 a week for one year, or 52 weeks. How much does he 
still need ? ; Ans, $252. 
15. A grocer, having 3 boxes of tea holding 80 pounds 
each, repacks it in six-pound boxes. How many boxes will 
it fill? Ans. 40 boxes. 
16. Two men go from the same point in opposite direc- 
tions, one 18 miles a day, the other 23. When they are 869 


miles apart, how many days have they travelled? 


How far apart are they at the end of one day? How many times is this 
number contained in 369? 


MULTIPLICATION, AND DIVISION. 65 


17. A company building a railroad, pay $377235 for labor, 
and $147690 for other expenses. The road is 15 miles long; 
what is the cost per mile? Ans. $384995. 

18. A farmer has 917 sheep in one field; 189 in another; 
276 in a third; and 379 in a fourth.’ If he divides them into 
three equal lots, how many will there be in each?. Ans. 587. 

19. The Vice-President’s salary is $8000 a year. If he 
spends $15 a day, how much will he lay up during his four 
years’ term, allowing 365 days to the year? Ans. $10100. 

20. Two men go from the same point in the same direc- 
tion, one at the rate of 41 miles a day, and the other 24. 
When they are 493 miles apart, how many days have they 


travelled? Ans. 29 days. 
21. I buy 29 horses for $1885, and sell them at a profit of 
$299. How much do I get apiece? Ans. $75. 


22. A planter raises 25963 pounds of cotton, and buys up 
35409 pounds more. If he packs the whole in bales of 458 
pounds, how many bales will he have? Ans. 134 bales. 

23. Roger Bacon, the inventor of spectacles, lived to the 
age of 78, and died 350 years before Sir Isaac Newton was 
born. If Newton was born in 1642, what was the year of 
Roger Bacon’s birth? Ans. 1214. 

= 24. How many hogsheads holding 1295 pounds each will 
it take to contain 76405 pounds of sugar? Ans. 59 hogsheads. 

25. Five partners make $1500 by a speculation. One of 
them divides his share between his two daughters. What 
does each daughter get? Ans. $150. 

26. Find the value of 637+ 8465367—487. Ans. 8465517. 

27. How much is 986743+9767+98437? Ans. 1094947. 

98. Find the value of 64372159—8524384. Ans. 55847775. 

29. Find the product of 79, 86, and 54. Ans. 866876. 

30. Divide 1824962163 by 26837548. Ans. 68, 8899 rem. 


66 FRACTIONS. 


FRACTIONS. 


103. When a whole is divided into 
two equal parts, each of these parts is 
called one Half. 


When a whole is divided into three 
equal parts, one of these parts is called 
one Third; two are two Thirds; &ce. 


When a whole is divided into fort 
equal parts, one of these parts is called 
= one lourth (or Quarter); two are called 

two Fourths; three, three Fourths; &c. 


In the same way we get Fifths, Sixths, Sevenths, 
&c., by dividing a whole into jiwe, sia, seven, &e., 
equal parts. The name is taken from the number 
of equal parts into which the whole is divided. 


104. The value of the part varies according to 
the number of parts into which the whole is divided. 
The more parts it is divided into, the smaller they 
must be. 


Half i Half 


Third ' Third 1 Third 


Fourth ' Fourth ' Fourth i Fourth 


One half of a thing is greater than one third; one third is 
greater than one fourth. 


103. When a whole is divided into equal parts, if there are two, what is 
each called? If there are three, what is each called? What are two called? 
If there are four equal parts, what is each called? What are three such 
parts called? How do we get Fifths, Sixths, Sevenths, &c.? From what is 
the name taken?—104. According to what does the value of the part vary? 
Which is greater, one half or one third? One fourth or one third? 


HOW WRITTEN. 67 


105. These equal parts into which a whole is 
divided, are called Fractions. 


Writing of Fractioms. 


106. Learn how fractions are written. 


One half 4 | Five thirteenths py 
One third x | Three twenty-seconds ik 
One fourth (quarter) t| Twenty sixty-firsts 3 
One tenth zo | Nine three-hundredths 7%; 
One two-hundredth 43>] Three thousandths To5T 
One thousandth 4/55 | Six twelve-hundredths 72°53 


107. A fraction, therefore, when written, consists 
of two numbers, one below the other, with a line 
between. 

The number below the line is called the Denomi- 
nator, It shows into how many equal parts the 
whole is divided, and therefore gives name to these 
parts. 

The number above the line is called the Numera- 
tor. It shows how many of the equal parts See 
by the Denominator are taken. 

The Numerator and the Denominator, taken to- 
gether, are called the Terms of the fraction. 


é is a fraction. 5 and 6 are its Terms. 6 is the Denomi- 
nator, and shows that the whole is divided into siz equal 


105. What are such equal parts of a whole called ?—106. Write one half; 
one third; one quarter; &c.—107. Of what does a written fraction consist ? 
What is the number below the line called? What does the Denominator 
show? ‘What is the number above the line called? What does the Numer- 
ator show? What are Numerator and Denominator, taken together, called ? 
Name the terms of the fraction jive sixths. Name its numerator; its de- 
nominator. 


68 FRACTIONS. 


parts, making each part one sixth. 5 is the Numerator, and 
shows that jive of these equal parts are taken. | 

In reading, name the numerator first,—jive siaths. Always 
pronounce ¢/ distinctly, in naming the part denoted by the 
denominator,—sivTu. 


- EXEROISE. 


Read these fractions. Then name the numerator and the 
denominator, and tell what each shows. 
5 . 237. Js 


et a py ae is ae 
O37 <b E Ord 2 3.G51 8 5x20 


ee ee ie | 
029 “3 0%.0 9 7 Ow oe 


Write the following fractions in figures :— 


1. Seven sixteenths. 8. Five seventy-seconds. 

2. Nine tenths. 9. Twenty-one ninetieths. 

3. Two five-hundredths. 10. Sixty ten-thousandths, 

4, Eleven billionths. 11. Sixteen twelve-hundred- 

5. Eighty’ hundredths. and-ninety-firsts. 

6. Three millionths. 12. Five hundred four-hun- 

7. One thousandth. dred-thousandths. 
Definitions. 


108, A Fraction is one or more of the equal parts 
into which a whole is divided; as, 4, 2. 

109. A Proper Fraction is one whose numerator 
is less than its denominator; as, 2, 33. 

110. An Improper Fraction is one whose numera- 
tor is equal to or greater than its denominator ; 
as, 3, 34. 

111, ah Mixed Number is onc that consists of a 
whole number and a fraction; as, 74 (seven and 


a half). 


In reading, which of the terms must be named first?—108. What is a - 


Fraction ?—109. What is a Proper Fraction ?—110. What is an Improper 
Faction ?—111. What is a Mixed Number? 


nF 2 ‘? 
‘3 ae 


avs 


TAKING FRACTIONAL PARTS. 69 


Fractional Parts of Whole Numbers. 


112, A fraction indicates division. The fractional 
line is the line used in the sign of division +. The 
dividend is written in place of the dot above the 
line; the divisor, in place of the dot below it. 
Hence, 


To find 3, divide by 2.| To find 1, divide by 4. 
To find 1, divide by 3.| To find 3, divide by 5. 


And generally, Zo jind one of the equal parts de- 
noted by a denominator, divide by the denominator. 


MENTAL EXEROISES. 


. How much is } of 14? Of 8? Of 18? Of 2? 

. How much is } of 20? Of 45? Of 10? Of 50? 

. Find + of 21. Of 56. Of 63. Of 14. Of 385. 

. Find 3 of 81. Of 18. Of 63. Of 27. Of 54. 

- qs of 802 Of 20? Of 100? Of 70? Of 30? 

- qs of 72? Of 48? Of 24? Of 12? Of 84? 

.% of 162 Of 80? Of 48? Of 24? Of 72? 

- 7 of 22? Of 662 Of 99? Of 33? Of 55? 

of 15? 4 of 542 } of 28? +, of 36? | of 36? 

of 24? § of 48? 4 of 27? +4 of 18? 54 of 11? 

of 830? 4 of 9? } of 162 § of 24? 3 of 12? 

1 of 50? ;3,; of 500? 5,55 of 4000? 1, of 640? 
1 of 6? +3 of 82? 1 of 15? 1} of 3? 

5 of 42? J of 20? of 77? of 56? 54 of 96? 


COMA P wd wy 


HS eS eS et 
Pe Sree he oe 

eed A sd 

S| 

io) 

h 

~y 

(=) 

ro) 


112. What does a fraction indicate? "With what does the fractional line 
correspond? Where is the dividend written, and where the divisor? How 
can we find one half? One third? Give the rule for finding one of the 
equal parts denoted by’a denominator 


‘@ 


70 FRACTIONS. 


113. Zo jind more than one of the equal parts 


into which a whole is divided, first find one in the. 


way just shown ; then multiply this by the number 
of parts required. 

Exampies.—1. How much is ? of 40? 

One fifth of 40 is 8; and three fifths are 3 times 8, or 24. 
Answer, 24. | 

9. A boy having 27 cents earned $ as much more. How 
many cents did he earn? 

He earned 4 of 27 cents. One ninth of 27 cents is 3 cents; 
and four ninths are 4 times 3 cents, or 12 cents. Answer, 
12 cents. 

MENTAL EXEROISES. 


1. How much is 2 of 18? # of 21? j5 of 108? 4% of 
108? <4 of 108? § of 42? 2 of 4? § of 86? §% of 20? 

2. How much is 55 of 22? ~& of 44? 3 of 72? 2 of 
40? 2 of 48? sof 30? 2 of 66? of 50? 3 of 64? 

3. How much is ;2, of 200? 3 of 272? #3 of 24? 3 of 
15? Zof18? Iof 81? 3 of 60? FL of 55? & of 72? 

4, A man having 48 cows sold 3 of them. How many did 
he sell? a 

5. At 81 cents a pound, what will § of a pound of tea cost? 

6. A boy having 20 cents gave away one tenth of them and 
spent two tenths. How many cents had he left? Ans. 14c. 


EXAMPLES FOR THE SLATE. 


1. Find =; of 280. Ans. 60. | 5. Find ;$, of 8410. Ans, 348. 
2. Find }} of 441. Ans. 99. | 6. Find ,}, of 8832. Ans, 128. 
3. Find 37 of 1610. Ans. 1242. | 7. Find =, of 7400. Ans. 275. 
4, Find 43 of 4200. Ans. 3010. | 8. Find ;57,5 of 6100000. 


118. How can we find more than one of the equal parts into whicha 
whole is divided? . 


REDUCING TO HALVES, ETC. 71 


114, A fraction indicates division. Hence, in 
division, when there is a remainder, in stead of 
writing it as such beside the quotient, it is usual to 
place it over the divisor in the form of a fraction. 


Exampre.—Find } of 45. 7) 45 
Answer, 63. 6, and 8 rem. 


115, Reducintig whole mumbers to halves, &c. 


2 halves make 1 whole. 5 fifths make 1 whole. 
8 thirds make 1 whole. 6 sixths make 1 whole. 
4 fourths make 1 whole. 7 sevenths make 1 whole. 


Hence, Zo reduce to halves, multiply by 2. 
To reduce to thirds, multiply by 8. 
To reduce to fourths, multiply by 4. 
To reduce to fifths, multyply by 5, &e. 


EXAMPLES FOR THE SLATE. 


1, How many thirds in 2901? Ans. 8703 thirds; or, 2423. 
2. If I cut 45 pies into halves, how many halves have 1? 

3. How many quarters of beef will 96 oxen make? 

4, How many tenths of an inch in 72 inches? 

5. Reduce 497 to fiftieths. Reduce 876 to twelfths. 

6. A cent is one hundredth of a dollar. How many cents 


are there in $75? Ans. 7500 cents. 
7, A mill is rooo Of a dollar. How many mills in $34? 

8. Find one eleventh of 25304618. Ans, 2300419;%. 

9. Find one thousandth of 19001. Ans. 19>55- 

10. Find one sixty-eighth of 53875. Ans. '794%5- 


114. What does a fraction indicate? Hence, in division, how is it usual 
to write a remainder ?—115. How do we reduce a whole number to halves? 
Tothirds? Tofifths? To twentieths% 


72 FRACTIONS. 


Reduction of Fractions, 


116: Reducing a fraction is changing its form 
without changing its value. 


If we divide a pie into 2 equal parts, 
__ each part is called 3. If we divide each 
half into 2 equal 
parts, wegetfour _| 
parts in all, and each is therefore 4 of * 
the whole. 

Now two of these fourths are made from one half, and 
are therefore equal to one half. 2, therefore, may be written 
4,—or, as it is generally expressed, reduced to 4. 


Reduction to Lowest ‘Terms. 


117. We have just seen that 2 = 1. Now what operation 
performed on ? gives}? Dividing both nu- 


Be 8 

merator and denominator ne 2. 4-2 —~ 2 
We have just seen that } = 37. Now what operation per- 
formed on } gives 2? Aaltipiaie s both nu- 1 ce 
merator and denominator by 2. a Daepenea 


The value of a fraction, therefore, rs not changed 
by dividing or multiplying both numerator and 
denominator by the same number. 


118. A fraction is said to be 7% zts lowest terms 
when its numerator and denominator are as small 
as they can be made without changing the value of 
the fraction. 1 is in its lowest terms; 2 is not. 


116. What is meant by Reducing a fraction? To what may 2 bereduced ? 
How do you know that 2 and 3 are equal ?—117. What operation performed 
on 2 givesi? What operation performed on } gives 2?° By what, then, is the 
value of a fraction not changed {—118. When isa fraction in its lowest terms ? 


REDUCING TO LOWEST TERMS. ‘@ 


ExAMPLE.—Reduce §° to its lowest terms. 


Divide both terms by 10, which will make 
them lower, while it will not change the 
value of the fraction. 

Looking at the resulting fraction § , we see that its terms 
can be divided by 3, which will make them 
lower without changing its value. Dividing, 
we get 2. . 

As we can find no number greater than 1 that will exactly 
divide the terms of this fraction, we know that it is in its 
lowest terms. Answer, 3. 

One division may reduce a fraction to its lowest terms, or 
several may be needed. 


GA esae? 6 
10[$2 = £ 


119. Rure.—TZo reduce a fraction to its lowest 
terms, divide its numerator and denominator by any 
number greater than 1 that 7s exactly contained in 
both. Dwide the result in the same way, repeating 
the process tull no number greater than 1 is exactly 
contained in both. 


EXAMPLES FOR THE SLATE. 


Reduce the following fractions to their lowest terms :— 


Ce ers Cie ae Oe oO eg ae 

2. Reduce 24 Ans. 2. | 9. Reduce ;§. Ans. 7. 
3. Reduce 3}. Ans. }. | 10. Reduce ;%%. Ans. 3. 
4, Reduce 33 Ans. }. | 11. Reduce 333. Ans. =. 
5. Reduce ;34. . Ans. 3. | 12. Reduce 4%. Ans. #5. 
6. Reduce 5. Ans, 3. | 18. Reduce 3%. Ans. 3's. 
7. Reduce ,°5 Ans. 3, | 14. Reduce 32. Ans. 2. 
8. Reduce 793. Ans. 7. | 15. Reduce }?3. Ans. 3. 


Reduce $9 to its lowest terms. How many divisions are needed, to re- 
duce a fraction to its lowest terres?—119, Give the rule for reducing a frac- 
tion to its lowest terms. 8 


74 FRACTIONS. 


Reducing Improper Fractions. 


120. An Improper Fraction is one whose numer- 
ator is equal to or greater than its denominator. 

121. An improper fraction may be reduced to a 
whole or mixed number. 


ExampLes.—l. Reduce 13 to a whole number. 

9 ninths make 1 whole; hence in 18 ninths there are as 
many wholes as 9 ninths are contained times in 18 br or 
2. Answer, 2. 


2. Reduce 2° to a mixed number. 

9 ninths make 1 whole; hence in 20 ninths there are as 
many wholes as 9 ninths are contained times in 20 ninths, or 
22. Answer, 22. 

In both these examples, it will be seen, the numerator is 
divided by the denominator. Hence the rule. ; 


122. Rute.—TZo reduce an improper fraction to 
a whole or mixed number, divide the numerator by 
the denominator. 


When the numerator is equal to the denominator, the 
value of the fractionis1. 3=1. 

When the numerator is greater than the denominator, the 
value of the fraction is greater than 1. 4=11. 

When, on dividing, a remainder occurs, it gat be placed 
over the canes in the form of a fraction. The answer 
is then a mixed number, as in Example 2. 

The fraction thus formed, if not already in its lowest 
terms, must be reduced. 


120. What is an Improper Fraction ?—121. To what may an improper — 
fraction be reduced? Give examples.—122. Give the rule for reducing an 
improper fraction to a whole or mixed number. When is the value of the 
fraction 1? When is it greater than1? When, on dividing, a remainder oc- 
curs, what must be done? If this fraction is not already in its lowest terms, 
what must be done? 


REDUCING IMPROPER FRACTIONS. 75 


ExAmpLE.—Reduce $3 to a whole or mixed number. — 


Divide the numerator 26) 40(1 Quotient. 
by the denominator. 26 114 
14. 
Reduce 14 to its lowest terms. 
2)i4 = J. Answer, 1,7. 


EXAMPLES FOR THE SLATE. 


Reduce these fractions to whole or mixed numbers :— 


i Ge elie Ge Si oie aia 2 

2. Reduce 3. Ans. 134. | 8. Reduce 3354. Ans. 11,4. 
3. Reduce 32. Ans. 33. | 9. Reduce %%. Ans. 255% 
4, Reduce #3. Ans. 7;4;. | 10. Reduce 898. <Ans. 202. 
5. Reduce 174. Ans. 5}. | 11. Reduce %%8. Ans. 873; 
6. Reduce 15°. Ans. 9,%. | 12. Reduce 73°. Ans. 863. 
7. Reduce 22°. Ans. 60. | 18. Reduce 222. Ans. 87. 


Reducing Mixed Numbers to Improper 
Fractions, 


123. A mixed number may be reduced to an 
improper fraction. 

Reduce 12} to an improper fraction. 121 
In 1 there are 4 fourths; and in 12, twelve 4. 
times 4 fourths, or 48 fourths. 48 fourths 4g +1=49 
and 1 fourth make 49 fourths. Ans. 42. 


Hence the following rule :— Ans, “7 


124, Rute.—To reduce a mixed number to an 
improper fraction, multiply the whole number by 
the denominator of the fraction, add in the nu- 
merator, and set the result over the denominator. 


123. To what may a mixed number be reduced? Give an example.— 
124, What is the rule for reducing a mixed number to an improper fraction ? 


76 FRACTIONS. i 


125. We have seen, in §115, how to reduce a whole num- 
ber to an improper fraction. The process is the same as that 
just shown, except that there is no numerator to be added in. 

A whole number may also be reduced to a fractional form 
by giving it 1 for a denominator. 8=? 7 = 7, 


EXAMPLES FOR THE SLATE, 


1. Reduce 8§ to an improper fraction. Ans, %. 
2. Reduce 4,4. 6. Reduce 1874. | 

8. Reduce 202. 7. Reduce 29,4. 

4. Reduce 89}. 8. Reduce 473%. 

5. Reduce 25}. 9. Reduce 152. 

10. Reduce 20 to a fractional form. Ans. 22, 
11. Reduce 15 to ninetieths. Ans, 135°, 


12. How many quarters in 52 yards? 

13. How many twelfths in 4854? In 525? 
14, How many nineteenths in 43? In 8,3 
15. Reduce 100;4, to an improper fraction. 
16. Reduce 999; to an improper fraction. 
17. Reduce 6403, to an improper fraction. 


Reducing to 2 Common Denominator. 


126. Fractions are said to have @ common de- 
nominator, when their denominators are the same. 
2 and 4 have’a common denominator. 

127. Fractions whose denominators are different 
may be reduced to others having a common de- 
nominator. 


125. How may a whole number be reduced fo an improper fraction? In 
what other way may a whole number be reduced to a fractional form? To 
what whole number is 2 equivalent ?—126. When are fractions said to have 
a common denominator ?—127, What may be done with fractions whose de- 
nominators are different ? 


COMMON DENOMINATOR. 44 


Exampie.—Reduce 2, 3, and 4, to fractions that 
have a common denominator. 


The three denominators are 5, 4, and 7%, Now, the 
product of three factors is the same, in whatever order they 
are taken. Hence, if we mul- 


tiply each denominator by the 5 x4 x t - ==. 140 
pp | 


other two, we shall get the 4x5x7=140 ache 


same number, and this willbe %7x5x4= 140 
the common denominator. 


¥ 


But the value of the fractions 


tA c 
Eo ee Fie = must not be changed. We must, 
ee ae therefore, multiply each numera- 
4 x5x1 140 tor by the same multipliers as its 
6 . ; E : — +94 denominator. Hence the follow- 


ing rule :— 


128. Rute.— Zo reduce fractions to others having 
a common denominator, multiply both terms of cach 
Sraction by all the denominators except its own. 


129. Whole numbers must first be reduced to a fractional 
form, and mixed numbers to improper fractions, 


EXAMPLES FOR WHE SLATE. 


Reduce the following fractions to equivalent ones having 
@ common denominator :— 


1. Reduce 2 and %. Ans. 23, 27. 
2, Reduce 3, 3, and ¢ Ang, 43, 595°24, 
3. Reduce ;%,, 2, and 3. Ans. 389%, $29, 449, 
4. Reduce 3, j, and ;{- Ans. $83) 16%) 182 
5. Reduce 3, ;4, and 3. Ang, 260 078, 
6. Reduce ;4, 44, and 6. Ang, POE e yee 


With the given example, show how fractions may be reduced to others 
having a common denominator.—128. Give the rule.—129. To what must 
whole numbers first be reduced? To what must mixed numbers first be 
reduced ?- 


78 FRACTIONS. | 


7. Reduce 3, 1, 7, and 4, to fractions having a common 


denominator. Ans. 33%, sf 28%) 286. 
8. Reduce oe J; ce 4. Ans. T30) 13 130) Tso 
0. Reduce #, $, 13. Ans. 225) $28, 984. 
10. Reduce +3 and 3% Ans. 343, 230. 
11. Reduce 3, 2, 2, 5. Ans, 22 ese eae. 
12, Reduce 3, 23, 3, 3 Ans Sao Foo) 350) Sse" 
13. Reduce 54, 2, and 2. Ans. #55, 299, 188. 
14. Reduce 4 and 34. Ans. $34, $47. 
15. Reduce 3}, 4, and 23 Ans, 3o8s ig Ze. 


A 


A@d@ition of Fractions. 


130. Halves and halves, thirds and thirds, &ce., 
can be added, just as we add pears and pears, 
dollars and dollars. 


EXAMPLE.—What is the sum of 5 halves : 
and 4 halves? Answer, 9 halves. a. ae 


Here the denominators are the same, and we simply add 
the numerators, and place the sum over the common de- 
nominator. 


1. Add =, 75, and +5 5 
2. Add 4, 2, and $%. 6. 
3. Add 3, 3, and ~%. fe 

I 8 


131. Halves and thirds, halves and fourths, &c., 
can not be thus dzrectly added, any more than we 
can add pears and dollars. They are things of 
different kinds. 


180. Can halves and halves, thirds and thirds, &c., be added? In what 
way? Give an example.—131. Can halves and thirds, halves and fourths, 
&c., be added directly? Why not? 


ADDITION OF FRACTIONS. 79 


Exampie.—What is the sum of 5 thirds and 3 halves? 


The parts being of different value, we can not put them 
together, and say they make 8 halves or 8 thirds. But, if we 
reduce them to parts of the same kind or value, we can then 
add them. 

By reducing to a common de- 
nominator, we find that 5 thirds are 
equal to 10 sixths, and 3 halves to9 19 ti 
sixths. 10 sixths and 9sixthsmake ° 
19 sixths. +42 being an improper 
fraction, we reduce it to 3}. Answer, 3}. 


—_ 


10 
6 


I! fl 
4 
4 oo 
=) 


ol 


— 
{SO BO poled eoftr 


Ans, 


o| 
Oo 
|— 


182. Rutz.—1. To add fractions, when they have 
a common denominator, add their numerators, and 
place the sum over the common denominator. 

When they have not a common denominator, re- 
duce them to fractions that have, and then proceed 
as above. 

Lf the resulting fraction is not in cts lowest terms, 
reduce tt. If itis an vmproper fraction, reduce tt 
to a whole or mixed number. 

2. Toadd mixed numbers, or fractions and whole 
numbers together, find the sum of the fractions sepa- 
rately, and add it to the sum of the whole numbers. 


Exampre.—Add together 3, 41, 2, and 5. 


Add the fractions: $4+i42 = 123, 
Add the whole numbers: 4+5 = 9. 
Add these two sums: 4° 

Ans. 1023 


a 


Show how we add § and 3.—132. Give the rule for adding fractions. 
Give the rule for adding mixed numbers, or fractions and whole numbers. 
Apply this latter rule in the given example. 


80 FRACTIONS. 


EXAMPLES FOR THE SLATE. 


1. Find the sum of } and 3. Ans. 155. 
2. Find the sum of 7 and $. Ans. 44. 
8. Find the sum of 3, and 3. Ans. 13%. 
4, Add together 34, 12, and 7. Ans. 231, 
5. Add together # and 25%. Ans, 2221, 
6. Add together 2, ~4, and ;%. Ans. 33%. 
7. Add together +4, 10, and 87. Ans. 18344, 
8. Add together 74, 3, 4, and $. Ans, 1223. 
9. Add together 8, 41, 2, and 1}. Ans. 9,5. 
10. Add together 3, 2, 3, and 23. Ans, 433. 


11. A hackman earns $2! one day, $34 the next, $4 the 
next, and $5} the next. How much does he earn in all four 


days? Ans. $15. 
12. How much land is there in 8 fields, containing 145, 
7;, and 232 acres? Ans, 45 acres. 


13. If I buy $2? worth of paper, and $6} worth of books, 
and give the storekeeper a ten-dollar bill, how much change 
will I receive? ; Ans. $1. 

14. Three men, buying a meadow, put in respectively 
$307,, $252, and $19,4. What does the meadow cost? 

15. A five-pound jar contains 32 pounds of bread and 24 
pounds of cake; what does the whole weigh? 

16. A peddler walks 8} miles one day, 5} the next, 104 
the next, and 12 the next; how far does he walk in all? 

17. How many pecks of peaches in four baskets, contain- 
ing respectively 24, 34, 23, and 3} pecks? 

18. If 5§ gallons of brandy are mixed with 1,°, gallons of 
water, how many gallons are there of the mixture? 

19. A lady hires a gardener for 15 cents an hour. How 
much must she pay him, if he works 6,; hours the first day, 
72 the second, and 53 the third? | Ans. 800 cents. 


SUBTRACTION OF FRACTIONS. 81 


” Subtraction of Fractions. 


133. The same principle applies in subtracting, 
'as in adding, fractions. Before subtracting, the 
parts must be made of the same kind or value, if 
they are not already so; that is, the denominators 
must be made the same. 


Examptes.—l. Frombdhalvestake4halves. , , 
Answer, 1 half. Sree 


2. From 5 thirds take 8 halves. 

Thirds and halves being parts of dif- 
ferent value, we must reduce them to 
parts of the same value. Reducing toa  ,, 
common denominator, we find that 5 thirds F 
are equal to 10 sixths, and 3 halves to 9 
sixths. 9 sixths from 10 sixths leave 1 sixth.. Answer, }. 


te 
to 
te 


i) 


o|— a0 oO} 


SO roles coftr 


e- Il Il Il 


be 
“ 


134. Rurtze.—1. Zo subtract one fraction from 
another, when they have a common denominator, 
take the numerator of the subtrahend from that of 
the minuend, and place the remainder over the com- 
mon denominator. 

When they have not a common denominator, re- 
duce them to fractions that have, and then proceed 
as above. 

Reduce the resulting fraction to rts lowest terms, 
or to a whole or mixed number, as may be necessary. 

2. Whole and mixed numbers may be reduced to 
improper fractions, before subtracting. 

133. What principle applies in subtracting fractions? Illustrate this, 


with the examples given.—134. Recite the rule. What may be done with 
whole and mixed numbers, before subtracting ? 


> e 


82 


FRACTIONS. 


EXAMPLES FOR THE SLATE. 


. 
1. From $ take 7. 4, From 12 take 4%. 

2, From i take 3. 5. From 23, take 74>: 
8. Take 545 from 3}. 6. Take 2% from #3. 

7, Subtract 2 from 33. Ans. sy. 
8. From ;3; saptnoes te Ans. 3; 
9. Subtract ~, from 4. Ans. 155. 

10. From 32 subtract 12. “Ans. 5. 

11. Take ~, from 2. 14, From % take 7. 

12. Take ;2, from 3. 15. From 7 take ? 

13. Take 2 from 4. 16. From 32 take 7%. 

17. From 1 subtract 2. Ans, }. 
[Reduce 1 to thirds; then proceed as before.] 

18. From 1 take 17. From 1 take %. 

19. From 4 subtract 2. Ans. 31. 
[Take 1 of the 4 avai and reduce it 4—=82 Min. 
to thirds. Then subtract the frac- _% Sub. 
tion. % from ? leaves }. Ans. 3}. 81 Rem.] 

20. From 5 siiiteaatta ws ae = 47] Ans, 44. 

21. From 17 subtract $ Ans. 1655. 

22. Take ;, from 3. Take js from 3. “Take 2 2 from 2. 

23. Subtract + from 53. Abs 5S. 


24, 
25. 
26. 


27. 
28. 
29. 
30. 


[Take 1 from ?, and annex the result to 5.] 
Subtract 4 from 24. From 23. Take 2 from 6% 
Subtract 3 from 62. Ans. 614, 
Subtract 2! from 4}. Ans. 122, 
[1 is greater than}. Hence reduce both mixed num- 
bers to improper fractions. The sum then becomes, 
Subtract 4 from 2°. Now proceed as before. ] 


to] 


oO 


Subtract 1,3, from 3. Ans. 12. 
Subtract 204 from 243. Ans. 32 
Subtract 5°, from 12, Ans, 11;3,. 
Subtract 3 from 13. Ans, }. 


MULTIPLICATION OF FRACTIONS. 83 


31. Subtract 14 from 91. Ans. 735. 

32. A grocer, having mixed 144 pounds of tea with. 264 
pounds of a different kind, sold all the mixture but 11 pounds. 
How much did he sell? 

33. If a person owning two farms, one of 702 acres, and 
the other of 120;5, acres, sells 90,4, acres, how much land has 
he remaining ? Ans. 1004 acres. 

34. How much paper has a printer left, if he had on hand 
271 reams, and has used 7} reams for one job, and 64 reams 
for another? Ans, 133, reams. 


Multiplication of Fractions. 
Fraction x Wuorte NuMBER. 


135. To multiply any number of equal parts by 
2, we may either take twice as many such parts, or 
make each part twice as great. That is, we may 
double the nwmber of parts, or double their size. 


Exampre.—Multiply 2 by 2. 

Double the number of parts. Twice 2? is §. 

Or, double the size of the parts. A half is twice as great 
as a fourth. Hence, twice ? is 2. 

§ may be reduced to 3. The two answers agree. 

Now, in the first case, we multiplied the numerator by 2. 
In the second case, we divided the denominator by 2. The 
latter mode is better, because it brings the fraction at once in 
its lowest terms. Hence the rule. 


136. Rutz.—Zo multiply a fraction by a whole 
number, diide its denominator by the whole num- 
ber, of this can be done without a remainder ; if 
not, multiply rts numerator. 


135. To multiply any number of equal parts by 2, what may we do? 
Illustrate these two methods, in multiplying % by 2. Which method is 
better ?—136. Give the rule for multiplying a fraction by a whole number. 


84 FRACTIONS. 


EXxAMPLES.—1. Multiply ,3, by 4. 
16 can be divided by 4. Divide it. Answer, 2. 
2. Multiply =; by 4. 
17 can not be divided by 4 without a remainder. 
Multiply the numerator. Answer, +2. 


137. Find the value of the following :— 


1.3.x 7. Ans. 42. 7. fA x ll. dns. 13. 
2. ee, Ans. 12. 8. # x 4, Ans. 42. 
3. 75 x 4. Ans. 23. 9. ~8, x 12 Ans. 1. 
4, 3 x 5. Ans. 3. 10. 754 x 12 Ans, 1,45. 
B. 2 x 6. ll. 45 x 8 

6. a x 6. 12. 4 x 10. 


Mixrep Numper x WHoLE NuMBER. 


138. Rute.— Zo multiply a mexed number by @ 
whole number, multiply the fractional part and the 
whole part of the mixed number separately, and add 
the products. 


Exampie.—Multiply 203, by 5. 


Multiply the fraction: 3x5=2=1! 
Multiply the whole part: 20x5—= 100 
Add the products: 1014 Ans. 
1. What will 10 clocks cost, at $54 each? Ans. $52. 
2. There are 51 yards in a rod; how many yards are 
there in 40 rods? .Ans. 220 yards. 


3. If 12 acres of land produce, on an average, 355 bushels 
of wheat each, what is the yield of the whole? 

4. A locomotive is moving at the rate of 257 miles an 
hour; how far will it go in 3 hours? Ans. 773 miles. 

5. What will 16 chairs cost, at $23? apiece ? 


138. Give the rule for multiplying a mixed number by a whole number. 


MULTIPLICATION OF FRACTIONS. 85 


Wrote Noumper x Fraction. 


139. Multiplying a number by a fraction is sim- 
ply taking such a part of it as is denoted by the 
fraction. 

Multiplying a number by } is taking } of it, or dividing 
it by 8, as shown in §112. Multiplying by 2 is taking } 
twice, or dividing by the denominator 3, and multiplying by 
the numerator 2, as shown in $113. 

In such cases, it is best to multiply first, as there may be 
a remainder on dividing. Hence the rule. 


140. Rute.— Zo multiply a whole number by a 
fraction, multiply it by the numerator of the frac- 
tion, and divide by the denominator. 


Examprte.—Multiply 140 by 2. 140 
Multiply 140 by the numerator 2, and 4 
divide the product by the denominator 3. 3 ) 280 
Answer, 933. 931 Ans. 
1. Multiply 160 by 4, 5713x8. Ans. 40803. 
2. Multiply 199 by 5. 8296x7. Ans. 64525. 
3. Multiply 200 by 6. 2644x119. Ans. 2403,4. 
7, A farmer owning 467 acres of land, sells 3? of it; how 


i ym ato 


many acres does he sell? Ans, 350} acres. 
8. How many pounds in 14 of a ton of coal, a ton being 
2000 pounds? Ans. 14663 pounds, 


9. How many pounds in § of a barrel of flour, there being 
196 pounds in a barrel? 

10. If a hogshead holds 63 gallons, and ; of its contents 
has leaked out, how much remains? Ans. 57 gallons. 


—. 


139. To what is multiplying a number by a fraction equivalent? To 
what, for instance, is multiplying by 4 equivalent? By #?—140. Give the 
rule for multiplying a whole number by a fraction. 


86 FRACTIONS. 


Wuoite Numever x Mixep Nvumser. 


141. Rutze.— Zo multiply a whole number by a 
mined number, multiply it first by the fractional 
part of the mixed number, then by the whole part, 
and add the products. 


EXAMPLE.—Multiply 317 by 63. ek 

Multiply 817 by 2: 2111 

Multiply 317 by 6: 1902, 

Add the products: 21131 Ans. 
1. Multiply 2463 by 73. Ans. 184723. 
2. Multiply 5698 by 63. Ans. 376064. 
8. Multiply 1275 by 394. Ans. 50065. 
4. Multiply 5416 by 853,. Ans, 4611723. 
5. Multiply 8570 by 73}. Ans. 6277525. 
6. Multiply 9087 by 2019. Ans. 19000044. 
7. Multiply 2639 by 62,5. Ans. 164633. 
8. Multiply 4471 by 49,7. Ans. 220920. 
9. Multiply 6381 by 564. Ans, 360172. 


Fraction <x FRActTIon. 


142. Multiplying by a fraction, we have seen, is 
equivalent to taking such a part as is denoted by 
the fraction. Multiplying 1 by } is equivalent to 
taking | of }. 

148, A fraction of a fraction, such as } of 4, is 
called a Compound Fraction. 

The same process is used in multiplying fractions 
and in reducing compound fractions to simple ones. 

141. Give the rule for multiplying a whole number by a mixed number. 


—142. To what is multiplying by a fraction equivalent ?—143. What is a frac- 
tion of a fraction called? In what two operations is the same process used ? 


ai OF FRACTIONS. 87 


144, Multip! yi by 2 


These fractions indicate aivielbi The numerators are 
the dividends. The denominators are the divisors. Multiply 
the numerators together to find the total dividend, and the 
denominators to find the total divisor. 

Then set the former product over the 1x #= 3 Ans. 
latter, in the form of a fraction. 


1. Multiply 2 by 7. Ans, 23, 
2. Multiply +, }, and 7 together. Ans. +13. 
3. Multiply 2, 4, and ;{ together. Ans. #3. 
4, Multiply 2, 4, 4, and } together. Ans. x5. 


5. Reduce 2 of } of 7 to a simple fraction. Ans. 3. 


145, CanceLtina.mWhen the same number ap- 
pears as a numerator and a denominator, draw a 
line through it, and omit it in multiplying. This 
is called “ee 


Ex.—Reduce a 2 of + of 3 to a simple fraction. 
; 3, 8 
3 an ae of 5 fo=a Ans. 


Cancel 8 and 8, 5 and 5. Then multiply the remaining 
numerators and denominators. Observe that 9 and 9 are not 
cancelled, because they are both denominators. 


1. Multiply 2 by § Ans. 4. 
2. Multiply 3, 3, ane together. Ans, 4. 
3. Multiply 7, 2, 74, and 3 together. Ans, 3. 
4. Reduce 3 of 2 of »° toasimple fraction. Ans. 18. 
5. Reduce 7 of ;4 of 3 toasimple fraction. Ans. }3. 
6. Reduce 5, of 2 of 2,1 to a simple fraction. 

7. Reduce £ of 2 of 7 of } to a simple fraction. 


144. In multiplying } by 3, show how we proceed, and why.—145. When 
the same number appears as a numerator and a denominator, what do we 
do? What is this called? Give an example. 


88 FRACTIONS. 


146. Equal factors in a numerator and a denomi- 
nator may also be cancelled. 


Ex.—Reduce z of § of 35 toa simple fraction. 
Throw the whole number 85 into a fractional form. 


2 5 

nt 8 Bp “~ = 62 Ans 

; of & of a = See 

3 

Cancel 8 (that is, divide by 3) in the first denominator 

and second numerator. Cancel 7 (that is, divide by 7) in the 
second denominator and third numerator. Then multiply 
the remaining factors. 


1. Multiply 3, 4, and % together. , Ans, 27. 
2. Multiply 4, § and uu together. Ans, 4. 
3. Multiply 3, 3, 3, and 2 Ene Ans, 27. 
4, Reduce ;4, of 7 of 33 of } Ans. 2%. 


147, Cancelling is dividing, 3 4.- 1 


When, therefore, a numerator 4 
or denominator disappears by a 1 (not 0) 
is left in its place. 


148. Cancelling is dividing. Hence, by cancelling before 
we multiply, we save the trouble of dividing after we multi- 
ply, tg reduce the result to its lowest terms. 


149, GenrraL Rute.—1. Zo multiply one frac- 
tion by another, or to reduce a compound fraction, 
Jjirst cancel factors common to any numerator and 
denominator; then multiply the numerators to- 
gether for a new numerator, and the denominators 
Jor a new denominator. 


146. In what other way may we cancel? Give an example.—147. When 
we cancel, what operation do we perform? When a numerator or denomi- 
nator disappears by cancelling, what is left?—148. What trouble do we 
avoid, by cancelling before we multiply ?—149. Give the general rule for the 
multiplication of fractions. 


MULTIPLICATION OF FRACTIONS. 89 


2. Whole numbers occurring in a compound frac- 
tion must first be reduced to a fractional form, and 
mixed numbers to improper fractions. 


EXAMPLES FOR THE SLATE. 


1. Multiply together ;%, §, 14, and 1%. Ans. 373, 


I 11° 

2. Multiply together 2, ?, }, and 15, Ans. 1. 

8. Multiply together 3, 3, 4, and }. Ans, =35. 

4, Find the product of 3, j%, and 312 Ans. xh 

5. Find the product of 34, 3%, 38, and 7. Ans. 34. 

6. Reduce % of 4 of 2 of 13. Ans. 17. 

7. Reduce 4 of 1 of 4 of 4. Ans.''=45 

8. Reduce 2 of 2 of 3 of 9. Ans. 1}. 

9. Reduce , of 2 of + of 4}. Ans. 44. 

10. Reduce § of 7 of 1° of 12. Ans. Ty. 


Mixrep Noumper x Mixep Noumper. 


150. Rutz.—Zo multiply two or more mixed 
numbers together, reduce them to improper fractions, 
and proceed as in multiplication of fractions. 

Exampre.—Multiply 42, 5,;, and 1} together. 

Reduce the mixed numbers to improper fractions. Then, 
cancelling, we get 41, or 354. 


14 71  uplaini b Mat ie 
Bes. aa X5= 5 a nA 
Find the value of the following :— 


1262x831. Ans. 981.| 4. 83x42x11, Ans. 17}. 
2. %ix1i. Ans. 181.| 5, 113x12xK12. Ans. 24%. 
8. 13x93, Ans. 183.| 6. 2}x82x52. Ans. 4411, 


150. Recite the rule for multiplying two or more mixed numbers together. 


90 FRACTIONS. 


SUMS FOR THE SLATE. 


1. A merchant owning 3? of a ship, sold ,7, of his share. 
What part of the ship did he sell? Ana. 21, 

2. What will 15} yards of velvet cost, at 4} dollars a 
yard? . Ans. $6733. 

3. A person having 4233 acres of land, left 2 of it to his 
son. What was the son’s share? Ans, 254 acres. 

4, A farmer has three wheat fields, of 44 acres each. 
Their average yield is 33? bushels to the acre. What is the 
yield of the whole? Ans, 425} bu. 

5. General Putnam lived to be 72 years old. Patrick 
Henry attained 3 of that age; how old was he at the time of 
his death ? 

6. How much flour must be laid in for a garrison of 355 - 
men, to allow each man 56} pounds? 

7. The British House of Commons contains 654 members. 
248 of this number are from England and Wales; how many 
does that make? Ans. 496 members. 

8. How many yards are there in a bale of linen, contain- 
ing 56 pieces, if there are 252 yards in each piece ? 

9. If a clock ticks sixty times in a minute, how many 
times will it tick in 152 hours, there being sixty minutes in 
an hour? Ans, 56160 times. 

10. If 680 persons subscribe for a work in three volumes, 
costing half a guinea a volume, what is the whole amount — 
of the subscription ? 

11. A owns 2 of a factory. He sells half his share to B, 
who in turn sells 3 of his share to C. What part of the fac- 
tory belongs to C? ANS. Pe. 

12. What will be the cost of three boxes of oranges, allow- 
ing 96 oranges to the box, at 13 cents apiece? 

13. Multiply 53, 54, and ,4 together. Ans. 1}. 


DIVISION OF FRACTIONS. 91 


Division of Fractioms. 
Fraction — Wuoite Number. 


151. To divide any number of equal parts by 2, 
we may either take half as many such parts, or 
make each part half as great. 

Exampre.—Divide ¢ by 2. 

Take half the number of parts. Half of 4 is 2. 

Or, make each part half as great. A tenth is half as great 
as a fifth. Hence half of is ;%. 

;; can be reduced to 7. ‘The two answers agree. 

Now, in the first case, we divided the numerator by 2. 
In the second case, we multiplied the denominator by 2. 
The former mode is better, because it brings the fraction at 
once in its lowest terms. Hence the following rule. 


152. Rurze.—1. Zo divide a fraction by a whole 
number, divide its numerator by the whole number, 
if this can be done without a remamnder ; if not, 
multiply its denominator. 

2. To dwide a mixed number by a whole number, 
reduce the mined number to an improper fraction ; 
then proceed as above. 


Exampres.—l. Divide 5, by 6. 

If the numerator 5 contained 6 exactly, we should divide 
it by 6.. As it does not, we multiply 
the denominator. BF El ame 

2. Divide 24 by 6. 

Reduce 24 to an improper frac- 
tion, 48. As the numerator 18 con- 
tains 6 exactly, divide it by 6.. 


151. To divide any number of equal parts by 2, what may we do? Iilus- 
trate these two methods, in dividing ¢ by 2. Which method is better ?—162. 
Give the rule for dividing a fraction by a whole number, 


92 - FRACTIONS. 


153. Find the value of the following :— 


1, 23+ 7. Ans. x. 8, $+ 9. Ans. +. 
2, 63 +9 9, a 

8, 121-211 (10. A +5. 

4, 7% +10. Ang, 3. 11. 13 + 18. Ans. 4+}. 
5, 22-4 12, 92 +3.) - 

6. 31 + 6 13, BL -+ 28, 

”, 51 +17 14. 55 + 40. 


Fraction —- FRActTIon. 


154. Divide 2 by 2 
_ That is, find how many times 2 is contained in 3. } is 

cepa a 1,7 times. Hence, in 2 it is contained 2 of 7 
times, or Cre: 

But 2 7 is twice as great as 1, and hence is contained only 
half as Bary times. 3 of 2,1 is 24. Answer, 24, or 2,5. 

Now, what have we done to the dividend 2, to produce 
21? We have multiplied its numerator by the denominator 
of the divisor 2, and multiplied its denominator by the nume- 
rator of the divisor. Or, in other words, 
we have inverted the divisor, andthen mul- 2X $= 
tiplied the fractions. Hence the rule. 


m|bo 
o\- 


155. Rurz.—1. To divide one fraction by another, 
multiply the dividend by the divisor inverted. 

2. Whole and mixed numbers must first be re- 
duced to improper fractions. 


Exampie.—Divide 41 by 21. 


Reduce the mixed numbers: , 2h gy 
Invert the divisor; cancel #1 3 Bae ® 3 
equal factors; multiply. 5 oe ee 1 Ans, 


154. Divide by 2. What have we done to the dividend 3, to produce 
this result ?—155. Give the rule for dividing one fraction by another. 


DIVISION OF FRACTIONS. 93 


EXAMPLES FOR THE SLATE. 


156. Find the value of the following :— 


1. ¢+ 3. Ans. 4. 7. 44 + 32 Ans, 1}. 

2. 35 + 55. Ans, 4. 8. 75 +12 Ans. 15. 

3. $+ +. Ans. 6. 9. 95 + 13. Ans. 65+. 

4. 6 + 2. Ans. 8. 10. 4% + 22. Ans, 23. 

5. 9+ 4. 11. 55 + 8, 

6.4+% 12, 11 + 38}, 

18. How many clocks, at $53 apiece, can be bought for 
$211? Ans. 4 clocks. 

14. If it takes 62 yards of lace to trim one dress, how 
many dresses will 60 yards trim? Ans, 9 dresses. 


15. A flock of sheep yield 104! pounds of wool. How 
many sheep are there, if they average 35 pounds of wool 


each ? Ans. 27 sheep. 
16. If a locomotive goes 1563 miles in 52 hours, what is 
its rate per hour? . Ans, 2721 miles. 


17. If a cow is allowed } of a bushel of turnips a day, 
how long will half a bushel last her ? 
18, Divide 834 by 21. Ans, 33. 


MISCELLANEOUS QUESTIONS ON Fractions.—What operation does 
a fraction indicate? What part of the fraction corresponds with the 
divisor? What corresponds with the dividend? What corresponds 
with the quotient? Ans. The value of the fraction. Which is 
greater, + or +? When we increase the denominator of a fraction, 
do we increase or diminish its value? Which is greater, + or #? 
When we increase the numerator of a fraction, do we increase or 
diminish its value? What kind of a fraction is $? What is its 
value? Is the value of a proper fraction greater or less than 1? 
Does multiplying a number by } increase or diminish it? Does 
dividing a number by } increase or diminish it? When we take 
+ of a number, do we multiply or divide by 4}? To what is can- 
celling equivalent? When can we cancel ? 


94 FRACTIONS. 


MISOELLANEOUS EXAMPLES. 


1. Find the sum, then the difference, then the product, of ~ 
1 and jy. Divide } by 7- 

2. If a boy’s wages are $4! a week, what will they 
amount to in 52 weeks? Ans. $221. 

3. Allowing 240 pins to a paper, how many pins are there 
in ;, of a paper? | , 

4, Three boys agreed to share their earnings for one week 
equally. The’ first earned $52; the second, $43; the third, 
$3. What was the share of each? Ans. $4335. 

5. A planter who has 56 hogsheads of sugar, sells ¢ of 
them to one merchant, and 3 to another. How many hogs- 
heads has he left? Ans, 21 hogsheads. 

6. If a man has to make a journey of 175;%, miles, how 
far will he have to go when he has travelled 48% miles? 

Ans. 1261 miles. 

7, How many quarter-dollars are there in $1250? 

8. A person owning 200 acres of land, leaves $ of it to 
his wife, and she divides her portion equally among her 5 
sons. What fraction of the whole does each son get, and 
how many acres? Ans. ;J5, 20 acres. 

9. If a merchant sells three dresses, of 10? yards each, 
from a piece of calico containing 40 yards, how many yards 
will he have left? Ans. 73 yards. 

10. Ifa farm of 148} acres is sold for $25751, what is the — 
price per acre? Ans. $17181, 

11. Three partners buy some silks for $1250, and sell 
them.for $18257. What profit has each? Ans. $25}. 

12. A tailor sold a coat for $201. If the materials cost 
him $92, and he paid $81 for making it, what was his profit? 

Ans. $233. 


FEDERAL MONEY. 95 


FEDERAL MONEY. 


157. Different countries have different currencies, 
or kinds of money. The currency of the United 
States is called Federal Money. 


Taste or Feprrat Money. 


10 mills (m.) make 1 cent,. . . ¢., ct. 


10 cents, J adime;s.* di 
10 dimes, 1 dollar,. . $ 
10 dollars, E eagle,...4. EB: 


158. All the denominations in this Table, except 
mills, are represented by coins. For convenience, 
other coins also have been issued. The coins of the 
United States are as follows :— 


Gotp. Double eagle, worth $20. 
73 


Eagle, $10. 
Half-eagle, Dera Os 
Three-dollar piece, “ §$ 3. 
Quarter-eagle,. «$28. 
’ Dollar, a1 ih a Ba 

Sitver. Dollar, Sear A Al ree re 
Half-dollar, & 50c. 
Quarter-dollar, vs 25. 
Dime, S 10¢. 
Half-dime, % 5c. 
Three-cent piece, “ 3c. 

Copper. Two-cent piece, ~ 2c. 
Cent, 5 Le. 


157. What is said of different countries? What is the currency of the 
United States called? Recite the Table of federal money.—158, What de- 
nomination of this table is not represented by a coin? Name the gold coins, 
and their value. The silver coins, The copper coins. 


96 FEDERAL MONEY. 


MENTAL EXEROISES. 

1. How many dollars are 5 double eagles worth? 

Moprr.—1 double eagle is worth $20; and 5 are worth 
5 times $20, or $100. Answer, $100. 

2, How many dollars are 8 half-eagles worth? 4 quarter- 
eagles? 6 eagles? 2 double eagles? 

3. How many cents in 2 half-dollars? In 4 quarter-dol- 
lars? In 7 dimes? In 3 half-dimes? 

4. How many dollars in an eagle and a half-eagle? In 
4 eagles and a half-eagle ? 

5. How many cents in 11 three-cent pieces? In 12 dimes 
and a half-dime ? 

6. How many eagles in 80 dollars? 

Mopret.—$10 make 1 eagle; in $80 there are as many © 
eagles as 10 is contained times in 80, or 8. Ans., 8 eagles. 

7. How many dimes in 50 cents? In 6 half-dimes? 

8. How many dollars in 18 half-dollars?. In 24 quarter- 
dollars? In 70 dimes? 

9. How many double eagles in $60? In $100? 


Writimg and Reading Federal Momey. 


159. In writing and reading federal money, the 
only denominations used are dollars, cents, and 
mills. 

160. Dollars are denoted by this sign $, always 
placed before the number.. They are separated 
from cents and mills by a point. The first two 
figures at the right of the point denote cents; the. 
third figure, mills: 

159. In writing and reading federal money, what denominations alone ~ 


are used? 160. How are dollars denoted? ‘Where are cents and mills 
found ? 


WRITING AND READING IT. 97 


161. Rurz.—TZo write federal money, set down 
’ the dollars with a point at the right. Set the cents 
in the first two places at the right of the point, and 
the mills in the third place. 

Lf the cents are expressed by one figure, fill the 
vacant place with a naught. If there are mills, but 
no cents, fill both vacant places with naughts. 


Ex. Six dollars, $6. 
Six dollars, fifty cents, $6.50 
Six dollars, fifty cents, one mill, $6.501 
Six dollars, five cents, one mill, $6.051 
Six dollars, one mill, $6.001 


162. Write the following: let the points range 
in line. 
1. Nine dollars, seventy cents. 
. Ninety dollars, five mills. 
. One hundred and forty dollars, seven cents. 
. Five dollars, seventy-five cents, seven mills. 
. Thirteen dollars, three cents, three mills. 
. Forty-one dollars, fourteen cents. 


co Ot HB CO WO 


163. Rutz.—/n federal money, read what rs at 
the left of the point as dollars, the first two figures 
at the right of the point as cents, and the third 
Figure as mills. 


164. Read the following :— 


$400.276 $350.70 $54. 
$112.009 $41.06 $6000.606 
$907.072 $1011.004 $789.001 


ee 


161. Recite the ry’ for writing federal money.—163. Recite the rule for 
reading federal money. 


98 FEDERAL MONEY. 


Addition of Federal Momey. 


165. Add $72.25, 374 cents, $9, and $15.625. 


$72.25 Here we are required to find the sum of sev- 
.o¢5 eral items in Federal Money. We must add 
9.00 things of the same kind. Therefore set dollars 
15.625 under dollars, cents under cents, &c., letting the 
$97,250 points all range in line. Represent the half-cent 
as 5 mills. Then add in the usual way, and set 
off dollars in the result by placing a point under the re 
in the items added. Answer, $97.25. 


166. Rute.—Write the several items, with their 
points ranging in line. Add, and place a point in 
the result under the points in the items added. 


Notr.—As there are no mills coined, less than 5 mills in 
a result are disregarded in business dealings, and 5 mills or 
more are counted as an additional cent. 


EXAMPLES FOR THE SLATE. 


Read the following expressions: find their sum, 


qd) (2) (3) (4) 
$849.75 $1969.454 $827.00 $27.50 
34.03 73.401 562.009 4,516 
460.983 1.011 437.09 875 
908.625 856.875 591.90 89.008 
376.009 28,652 875.63 43.921 


5. Bought a box of raisins for $1.82, a bushel of apples 
for 88 cents, a cheese for $5.94, and a barrel of sugar for 
$27.62. What did the whole amount to? 


165, Set down the givenexample. Howmust we place dollars, cents, &c. ? 
How do we represent the half-cent? How do we set off dollars in the result? 
—166. Give the rule for the addition of federal money. 


ADDITION OF FEDERAL MONEY. 99 


6. A farmer receives $15.87 for a cow, $75 for a horse, 
$3.13 for some potatoes, and $5.55 for some poultry. How 
much does he receive in all? . Ans. $99.05. 

7. Sold some velvet for $3.33, broadcloth for $18.75, silk 
for $12.50, muslin for $5.40, carpeting for $30.05, a shawl 
for $12.25. What is the amount of the bill? Ans. $82.28. 

8. If a house cost $3487.75; repairs, $53.87; painting, 
$119.23; furniture, $1563.39; moving, $9; what was the 
whole cost? Ans. $5282.74. 

9. A lady gives 25 cents for needles, $17.50 for a dress, 
$2.63 for trimmings, $1.50 for a cap, and 12 cents for thread. 
How much does she lay out? Ans. $22. 

10. A man lends $68 to one friend, $443.75 to another, 
and $19.05 to a third. How much does he Jend in all? 

11. Add together sixty dollars; five cents, six mills; sixty 
cents; six hundred and fifty dollars, five mills; four mills; 
fifty-nine cents. Ans. $711.255. 

12. There were taken up in a church collection, 16 cents, 
8 three-cent pieces, 10 half-dimes, 8 dimes, 4 quarter-dollars, 
8 half-dollars, and a two-dollar bill. What did the whole 
amount to? Ans. $6.05. 


Subtraction of Federal Momey. 


167, If a person owing $143 pays $27.37, how 
much does he still owe? 


$143.00 We are here required to find the difference 
97.37 between $148 and $27.37. Set the subtrahend 
$115.63 under the minuend, filling the vacant places of 
: the latter with naughts. Place dollars under 
dollars, cents under cents, &c. Subtract in the usual way, 
and set off dollars in the remainder by placing a point under 
the other points. Answer, $115.63. 


100 FEDERAL MONEY. 


168. Rurz.— Write the subtrahend under the 
minuend, with their points ranging m line. Sub- 
tract, and place a point in the remainder under the 
other points. 


EXAMPLES BOR THE SLATE. 


(1) (2) (8) (4) 

From $438.69 $101.467 $50.001 $64. 
Take 17.748 88.35 * 9.099 625 
5. From forty-six dollars, two mills, subtract eighteen 
cents, nine mills. Ans, $45,818. 
6. From one hundred dollars, three cents, take seven 
dollars, seven mills. Ans. $93,023. 


7, A person, having bought goods to the amount of 
$65.76, gave the storekeeper a hundred-dollar bill. How 
much change must he receive? Ans. $34.24. 
"8. If a merchant sells goods that cost him $151.82 for 
$99.99, does he gain or lose, and how much? 

9. Bought a cow for $37.25; paid on account $6.87. 
How much remains unpaid? 

10. Paid for a lot $947.25; for erecting a house, $2345.47; 
for furniture, $1159. Isold the whole for $4500. Did I gain 


or lose, and how much ? Ans. Gained $48.28. 
11. A man worth $10000 gave away $956.38, and lost 
$1127.82. What was he then worth? Ans. $7915.80. 


12. If a lady gives 12 cents for ink, 63 cents for pens, 
$13.30. for books, and $1.87 cents for paper, how much 
change must she get for a twenty-dollar bill? Ans. $4.08. 

13. Bought $75 worth of hay, and $25.25 worth of corn; 
paid $49.88. How much is still due? Ans. $50.87. 

14. From fifty dollars, take fifty cents, five mills. 


168, Give the rule for the subtraction of federal money. 


MULTIPLICATION OF FEDERAL MONEY. 101 


Multiplication of Federal Memey. 


169. What do 15 cows cost, at $16.345 each ? 


$16.345 If 1 cow costs $16.345, 15 cows cost 15 
15 times $16.345. Find the product, and point off 
81795 from the right three figures for cents and mills, 
16345 because there are three figures representing 
cents and mills in the multiplicand. Answer, 


245.175 — g045.175, 

170. Rure.— Multiply in the usual way; point 
of from the right of the product, for cents and 
malls, as many figures as represent cents and mills 
in the multiplicand. 


EXAMPLES FOR THE SLATE. 


(1) (2) (3) (4) 
Multiply $64.275 $15.89 $37.59 » $293.872 
By 9 12 13 56 


5. What will 42 calves cost, at $3.75 apiece ? 

6. At 374 cents apiece, what will 75 geese cost? 

7. What will 890 cords of wood cost, at $3.78 a cord? 

8. What will be the cost of 144 yards of black silk, at 


$1.20 a yard? Ans, $17.40. 
9. If a boy’s wages are $4.75 a week, how much will he 
earn in a year, or 52 weeks? Ans. $247. 
10. If a clerk earns $8 a week, and spends $4.75 a week, 
how much will he lay up in a year? Ans. $169. 
11. What will it cost six persons to board for a year, at 
the rate of $5 apiece each week? Ans. $1560. 


169. Go through the several steps in the example.—170. Recite the rule 
for the multiplication of federal money. 


102 FEDERAL MONEY. 


Division of Federal Money. 


171, If 4 piano-fortes cost $1501, how much do 
they cost apiece ? 

4.) S1501 If 4 pianos cost $1501, one piano will cost 
83754 one fourth of $1501. Dividing, we find the 
quotient to be $8754. 

If cents and mills are required in the 4) $1501.00 
answer, in stead of the fraction of a dollar, $375.95 
annex ciphers and continue the division. . ¢ 
Point off in the product, for cents and mills, as many figures 
as represent cents and mills in the dividend. Ans., $375.25, 

25 cents make a quarter of a dollar. The answers agree. 


172. Rute.—Divide in the usual way ; point off 
From the right of the quotient, for cents and mills, 
as many figures as represent cents and mills in the 
dividend. | 


EXAMPLES FOR THE SLATE. 


(1) (2) (3) (4) 
8) $48.816 9) $47.88 4) $106 7) $82.53 
G5 .ATT $ $ $11.79 


* 5. Divide $12.48 equally among 12 persons. 
6. If 86 hats cost $88.92, how much is that apiece? 
7. If a farmer sells 240 bushels of oats for $182.48, how 
much does he get a bushel? Ans. $.552. 
8. Find ;},; of $424.632. Ans. $4.083. 
9. Four partners bought some land for $1150. They sold 
it for $940 cash and $500 worth of grain. How much did 
each make by the bargain? Ans. $72.50, 


171. Divide $1501 by 4 in both the ways shown above.—172. Recite the 
rule for the division of federal money. 


MISCELLANEOUS EXAMPLES. 1038 


MISCELLANEOUS EXAMPLES IN FEDERAL MONEY. 


1. If a person spends $410.28 in a year, how much is that 
a week, allowing 52 weeks to a year? Ans. $7.89. 

2. A man buys 4 barrels of flour, at $5.95 a barrel; 18 
chickens, at 29 cents each; and 56 pounds of butter, at 274 
cents a pound. What does the whole cost? Ans. $44.42. 

Find the cost of each item ; then add the three amounts. 

3. Bought 8 pair of gloves, at $.75 a pair; 12 yards of 
lace, at $1.38 a yard; 10 yards of sheeting, at 45 cents a yard. 
What is the cost of the whole? Ans. $28.31. 

4, A lady buys 2 turkeys at $1.25 each, and 5 bushels of 
potatoes at 94 cents a bushel. How much change will she 
receive for a $20 bill? Ans. $12.80. 

5. A man pays for some land $400 cash and $192.80 in 
produce. If there were 57 acres, how much does the land 
cost him per acre? Ans. $10.40. 

6. If a boy buys 12 knives for $7.50, and sells them at 75 
cents apiece, how much does he make on each? Ans. $.125. 

Find the cost of 1 knife, and subtract it from the selling price. 

7. Divide $2117.71 equally among 35 families, and find 
the share of each. Ans. $60.506. 

8. Four persons contribute $8000 for a speculation. The 
first puts in $99.05; the second, $2460.80; the third, $986. 


What does the fourth put in? Ans. $4454.65. 
9. If 184 pounds of coffee are sold for $52.44, what is the 
rate per pound ? Ans. $.285. 


10. A father who has $2450 in stock, a house valued at 
$4750, and bonds to the amount of $15040, divides the whole 
equally among his two sons and three daughters. What is 
the share of each? Ans. $4448. 

11. If 40 acres of meadow land are worth $1260, what 
is 2 of the tract worth? Ans. $840. 


104 REDUCTION. 


REDUCTION. 


173, How many cents in five dollars ? 


In 1 dollar there are 100 cents, and in 5 dollars 5 times 
100 cents, or 500 cents. Answer, 500 cents. 

We have here changed the denomination from dollars to 
cents, without changing the value. This process is called 
Reduction. We have reduced dollars to cents. 

174, Reduction is the process of changing the 
denomination of a number without changing its 
value. 

175. There are two kinds of Reduction :— 

1. Reduction Descending, in which we change 2 
higher denomination to a lower, as dollars to cents. 
Here we must multiply. 

2. Reduction Ascending, in which we change a 
lower denomination to a higher, as cents to dollars. 
Here we must divide. 


Reduction Descendimg. 


176. Reduce $27 to mills. 


° 100 cents make $1; in $27, therefore, 


se 0 there are 100 times 27 cents, or (annexing 

ee two naughts) 2700 cents. 
2700 ec. There are 10 mills in 1 cent; in 2700 
10 cents, therefore, there are 10 times 2700 


27000 m. mills, or (annexing one naught) 27000 mills. 
Answer, 27000 mills. 


178. Solve the example given. What have we done in this example ?— 
174. What is Reduction ?—175. How many kinds of reduction are there? 
What do we do in Reduction Descending? What operation must we perform ? 
What do we do in Reduction Ascending? What operation must we perform? 
—176. Reduce $27 to mills, explaining the steps. 


REDUCTION DESCENDING. 105 


177, Reduce $27.465 to mills. 


Reduce $27 to cents: 27 x 100 = 2700 c. 

Add in 46 cents: ; 2700+46 = 2746 c. 
Reduce 2746 cents to mills: 2746x10— 27460 m. 

Add in 5 mills: 27460 +5 = 27465 m. Ans. 


Compare this result with $27.465, the amount to be re- 
duced. It is the same, with the dollar mark and point 
omitted. 


178. GenrRAL Rute ror Repucrion Drscenpine. 
—Multiply the highest gwen denomination by the 
number that vt takes of the next lower to make one 
of this higher, and add in the number belonging to 
such lower denomination, if any be given. 

Go on thus with each denomination in turn, till 
the one required is reached. 


179. Lederal Money.—In federal money, the 

above rule is applied as follows. See the examples 
in §176, 177. 

1. Loreduce dollars to mills, annex three naughts ; 
to reduce dollars to cents, two; to reduce cents to 
mills, one. 

2. To reduce dollars and cents to cents, or dollars, 
cents, and mills, to mills, simply remove the dollar 
mark and the pot. 


180. Reduce the following :— 


1. $624 to cents. 4, $450.63 to mills. 
2. $125 to mills. 5. $29.172 to mills. 
8. $.485 to mills. 6. $50000 to mills. 


177. Reduce $27.465 to mills, explaining the steps. How does the result 
compare with the original amount ?—178. Recite the general rule for reduc- 
tion erence .—179. Give the rules for the reduction of federal money, 

5 


106 REDUCTION. 


7. Reduce 5 eagles to mills, Ans. 50000 m. 
8. How many cents in 28 eagles? Ans. 28000 c. 
9. How many mills in 15 double eagles? 


10. How many half-dimes in 78 eagles? 

11. Reduce $450.59 to mills. 

12. How many dimes are $674 worth? 

13, A lady gets a half-eagle changed to dimes. How 


many dimes should she receive? Ans. 50 d. 
14, If a boy gets a $10 bill changed to half-dimes, how 
many should he receive? Ans. 200 half-dimes. 


Reduction Ascending. 
181. Reduce 27465 mills to dollars. 


10 mills make 1 cent; therefore in 27465 mills there are 
as many cents as 10 is contained times in 27465. Dividing 
by 10 (cutting off one figure), we get 


10 ) 27465 ae 2746 cents, and 5 mills over. 
100) 2746 ¢., 5m. 100 cents make 1 dollar; there- 
Ans. $27.465 fore in 2746 cents there are as many 


dollars as 100 is contained times in 
2746. Dividing by 100 (cutting off two figures), we get $27, 
and 46 cents over. Answer, $27, 46 cents, 5 mills; or, 
$27.465. 
Compare this result with 27465 mills, the amount to be 
reduced. We have simply pointed off three figures from the 
right, and inserted the dollar mark. 


182. In $177 we reduced $27.465, and obtained 
27465 mills. In $181 we reduced 27465 mills, 
and obtained $27.465. Thus it will be seen that 
Reduction Descending and Reduction Ascending 
prove each other. 

181. Reduce 27465 mills to dollars, explaining the steps. How does the 


result compare with the amount given to be reduced?—182, By comparing 
the examples in § 177 and 181, what do we find? 


REDUCTION ASCENDING. 107 


183. GenrraL Route ror Repuction AscEnpIna. 
—Diwide the gwen denomination by the number 
that wt takes of it to make one of the next higher. 
Dwwide the quotient in the same way, and go on 
thus till the required denomination rs reached. 
The last quotient and the different remainders form 
the answer. 


184, Lederal Money.—In federal money, the 
above rule is applied as follows :— 

To reduce mills to dollars, point off three figures 
Srom the right ; to reduce cents to dollars, two; to 
reduce mills to cents, one. 


EXAMPLES FOR THE SLATE. 


. Reduce 4790 mills to cents. Ans, 479 c. 
. Reduce 59195 mills to dollars. 

. Reduce 461063 cents to dollars. 

. How many dollars in 70000 mills? 

. How many dollars in 85310 cents? 

How many eagles are 50 gold dollars worth? 
. Reduce 2500 dimes to eagles. 

. How many dimes are equal to 600 mills? 

. Reduce 46000 cents to double-eagles. 

. Reduce 4676 mills to cents, 

. How many half-eagles in 1500 cents? 

. How many dollars in 200 half-dimes? 

. How many double eagles in 1200 half-dimes? 
. Reduce 1623487 cents to dollars. 

. How many mills in 5 three-dollar pieces? 

. How many cents in 4 quarter-eagles? 


OOD oP O&O DO 


jt tt pt 
a or WON KF OC 


183, Recite the general rule for reduction ascending.—184, Give the rule 
for the reduction of federal money, 


108 COMPOUND NUMBERS. 


COMPOUND NUMBERS. 


185. A Compound Number is one consisting of 
different denominations: as, 3 dollars, 14 cents; 
5 feet, 10 inches. 

186. To show the relations that different denomi- 
nations bear to each other, Tables are constructed. 
They must be learned perfectly. 


English or Sterling Momey. 


187. The currency of Great Britain is called 
English or Sterling Money. 


TABLE. : 
-4 farthings (far., qr.), 1 penny, .. d. 
12 pence, . 1 shilling, . s. 
20 shillings, 1 pound, .. 2: 
21 shillings, 1 guinea,. . guin. 


188, The pound mark always precedes the number; as, £2. 
Farthings are sometimes written as the fraction of a penny; 
2d.3far., or 23d.; 38d.2far., or 34d. 

The pound is simply a denomination. A gold coin called 
the Sovereign represents it. The Sovereign is worth $4.84, 
The English penny is worth about 2 of our cents. . 

* Guineas, originally made of gold brought from Guinea, are 
no longer coined. 

The Crown is a silver coin, worth 5 shillings. 

185, What is a Compound Number ?—187. What is the currency of Great 
Britain called? Recite the Table of Sterling Money.—188. How must the 
pound mark stand? How are farthings sometimes written? What repre- 


sents the pound? What is the sovereign worth? The penny? What is 
said of guineas? What is the crown worth? 


_ them to shillings, because 20 shillings 


STERLING MONEY. 109 


EXAMPLES FOR THE SLATE. 
189. Recite the rules for Reductioh, $178, 183. 


1. In £7 5s. 1 far. how many farthings? 


Multiply th 
ultiply the £7 by 20, to reduce £7 Bs. 1 far. 
20 


make a pound. Add in the 5 shillings. Sper 


Multiply 145s., thus obtained, by 12, ee 8. 
to reduce them to pence, because 12 Read 
pence make a shilling. There are no sia d. 


pence in the given number to add in. pene 
Multiply the 1740d., thus obtained, 6961 far. Ans. 

by 4, to reduce them to farthings, because 4 farthings make 

apenny. Addin the 1 farthing. Answer, 6961 far. 


2. Reduce 15383 far. to pounds, shillings, &c. 


4) 15383 far. Divide by 4, to reduce to pence. 
12) 38 45 3 far. Divide the quotient, 3845d., by 12, to 
reduce it to shillings. 


2/0) 82)0 5d. Divide the quotient, 320s. by 20, to 
£16 reduce it to pounds. The quotient and 

Ans, £16 53d. remainders form _the answer. Always 

mark the denominations throughout, as in these examples. 
3. Reduce £75 to pence. Ans. 18000d. 
4. Reduce 19s. 6d. to pence. Ans. 234d. 
5. Reduce 15s. 8d. 2 far. to farthings. Ans. 734 far. 
6. Reduce 8670d. to pounds, &c. Ans. £36 2s. 6d, 
7. Reduce 16255s. to pounds. Ans. £812 15s, 
8. Reduce 24681 far. to. guineas, &c. Ans. 24 guin., &e. 
9. Reduce £3 14s. 73d. to farthings. Ans. 8582 far. 


10. Reduce 1920 far. to pounds, &c. 

11. Reduce 3 guin. 10s, 6d. to farthings. 
12. Reduce 16s. 103d. to farthings. 

13. Reduce 1080 farthings to pounds, &c. 
14. Reduce 8628 pence to pounds, &c. 


110 COMPOUND NUMBERS. 


Troy Weight. 


190. Troy Weight is used in weighing gold, silver, 
precious stones, and liquors; also in philosophical 
experiments. 

TABLE. 
24 grains (gr.) make 1 pennyweight, pwt. 
20 pennyweights, 1 ounce,.... oz. 
12 ounces, 1 pound, saeneunam 


EXAMPLES FOR THE SLATE. 
1. Reduce 48494 gr. to pounds, &c. Ans. 8lb. 5 0z. 14 ger. 
Divide by 24; then by 20; then by 12. 


2. In 41b. 60z. 13 pwt. how many grains? Ans. 262382. 


Multiply 41b, by 12; add in 6. Multiply the result by 20; add in 
13. Multiply this result by 24. Mark the denominations through- 
out. 


. In 1001b. 1 gr. how many grains? Ans. 576001 gr. 
. Reduce 8976 pwt. to pounds. Ans. 87]b. 40z. 16 pwt. 
. Reduce 9 0z. 5 pwt. 20 gr. to grains. Ans, 4460 gr. 
. How many pounds, &c., in 1180 oz. ? 
. Reduce 161b. 10 0z. to pennyweights, 
. Reduce 5l1b. 5 pwti to pennyweights. 
. Reduce 121b. 1 oz. 9 pwt. to grains. © Ans, 69816 gr. 
10. If a miner digs 20 pwt. of gold one day, and 87 the 
next, how many ounces will he have? Ans. 20z. 17 pwt. 
, 11. How many pounds in two bars of silver, one of which 
weighs 240 pwt., and the other 860 pwt. ? 
12. How many pennyweights in 4 rubies, weighing 8 gr., 
10 gr., 12 gr., and 24 gr. ? Ans. 2pwt. 6 gr. 
13. A lady has 36 tablespoons weighing 34 pwt. each, and 
24 teaspoons weighing 19pwt. each. How many pounds do 
all her spoons weigh ? Ans. 7 1b. 


Oo MOT oD Ct Pp CC 


APOTHECARIES’ WEIGHT. 111 


Apothecaries’ Weight. 


191. Apothecaries’ Weight is used by apothecaries 
in mixing medicines. They buy and sell their 
drugs, in quantities, by Avoirdupois Weight, given 
on the next page. 


TABLE. 

20 grains (gr.) make 1 scruple, sc. or 9. 
3 scruples, Saran Ur Onr ae 
8 drams, 1 ounce, . .0z.or 3. 

12 ounces, 1 pound,. . lb. or fb. 


192, The ounce and pound of Apothecaries’ 
Weight are the same as in Troy Weight. 


MENTAL EXEROISES. 


. How many ounces in 7} pounds of senna? 

. Reduce 1 ounce to grains. 

. Reduce 94 ounces to drams. 

. Which is greater, 153 or 143? 

. How many grains in 23 of magnesia? 

. How many scruples in 94 drams? 

. How many doses of 12 grains each in 13 of musk? 

. How many powders of ten grains each can a druggist 
make out of 25 of calomel? 

9. Divide a dram of jalap into six powders; how many 
grains will there be in each? 

10. If a druggist sells four customers 6 0z. of blue vitriol 
each, out of a five-pound package, how many pounds has he 
left ? 

11. How many grains in the following mixture: nitrate 
of silver, 5gr.; opium, $3; camphor, 1D? 


OTS & PR oO RH 


112 COMPOUND NUMBERS. 


Avoirdupois Weight. 


193. Avoirdupois Weight is that In common use. 
By it are weighed all articles not named under 
Troy and Apothecaries’ Weight; such as groceries, 
meat, coal, cotton, and all the metals except gold 
and silver. 


TABLE. 
16 drams (dr.) make 1 ounce,..... . 02. 
16 ounces, 1 pound,” . i aes 
25 pounds, 1 quarter, .42%0n 
4 quarters, 1 hundred-weight, ewt. 


20 hundred-weight, 1 ton, ..... ..d5 


194. The ounce of Avoirdupois Weight is less 
than the Troy ounce, but its pound is greater than 
the Troy pound. 


195. It was formerly customary to allow 112 pounds to 
the hundred-weight, and 28 pounds to the quarter. But this 
is now seldom done, except in the case of coal, iron, and 
plaster bought in large’ quantities, and English goods passing 
through the Custom House. 

Twenty hundred-weight of 112 pounds make a ton of 
2240 pounds, which is distinguished as a Long Ton. 


MISCELLANEOUS QuEsTIONS.—In what denominations do British 
merchants keep their accounts? How much is a half-crown worth? 
What weight is used-in weighing emeralds? In weighing hay? 
Coins? Cotton? In weighing drugs for a physician’s prescription ? 
Recite the Table used in philosophical experiments, ° Recite the Table 
used in weighing cheese. Recite the Table used in mixing drugs. 
Which is greater, the Troy pound or Apothecaries’ pound? The 
Troy ounce or Avoirdupois ounce? The Apothecaries’ pound or 
Avoirdupois pound? How many pounds were formerly allowed to 
the hundred-weight? In what alone is this now done? What is a 
Long Ton? . 


AVOIRDUPOIS WEIGHT. YES 


EXAMPLES FOR THE SLATE. 


1. In 873450 drams, Avoirdupois Weight, how many tons, 
&e. ? Ans, 1 T. 14 ewt. 11 Ib. 14 oz. 10 dr. 


Does this example fall under Reduction Descending or Reduction Ascend- 
ing? Repeat the rule for Reduction Ascending. Name the numbers in order, 
by which we have to divide. In all examples in Reduction, be careful to mark 
the denominations throughout. 

Prove this example by reducing your answer to drams. If the result 
agrees with the number of drams given above, your work is right. 


2. Reduce 5ewt. 211b. 40z. to ounces. Ans. 8340 oz. 


Which kind of Reduction does this fall under? Repeat the rule for 
Reduction Descending. Name the numbers in order, by which we have to 
multiply. Why do we not first multiply by 20? As there are no quarters to 
add in, we may multiply the 5cwt. by 100 at once, to reduce them to pounds, 
in stead of by 4 and 25. 


3. Reduce 1T. lewt. 1dr. to drams. Ans. 537601 dr. 
4, Reduce 856702 drams to tons, hundred-weight, &c. 
Ans. 1T. 18 cwt. Iqr. 21]b. 7oz. 14dr. 
5. How many tons in 60000]1b. of lead? Ans. 80T. 
6. Reduce 8cwt. 15 0z. to drams. Ans. 77040 dr. 
7. How many pounds in 402 tons? Ans. 81500 |b. 
8. How many ounces in 6714 cwt.? 
9. Reduce 602000 oz. to tons. Ans. 18T. 16 cwt. 1 qr. 
10. What will 54 cwt. of poultry cost, at 9c. a pound? 
Reduce 5} cwt. to lb.; then multiply by the price of 1]b. 
11. What cost 233 cwt. of beef, at 12c. a lb.? Ans. $285. 
12. How many tons, &., in 18 bales of cotton, weighing 
550 Ib. apiece ? Ans. 3T, 1lcewt. 2 qr. 
13. How many pounds in 50 tons? In 50 long tons? 
14. If I buy 370 long tons of coal, and sell 370 ordinary 


tons, how many pounds have I left? Ans. 88800 Ib. 
15. Bought 7T. 18cwt. of iron. What does it cost, at 4 
cents a pound? Ans. $632. 


16. How many ounce balls can be moulded out of 25 
pounds of lead? 


114 COMPOUND NUMBERS. 


196. In connection with Avoirdupois Weight, 
learn the following 


MiscELLANEOUsS TABLE. 


14 pounds, ...1 stone of iron or lead. 
56 pounds,...1 firkin of butter. | 
100 pounds,...1 quintal of dried fish. 
196 pounds,...1 barrel of flour. 
200 pounds, ...1 bl. of beef, pork, or fish. 


EXAMPLES FOR THE SLATE. 


1. What is the cost of a firkin of butter, at 20 cents a 
pound? Ans. $11.20. 

2. How many packages of 7 pounds each can be made 
out of a barrel of flour? 

3. How many pounds in 424 stone? 

4. If a grocer who has 7 barrels of pork, sells half a 
barrel, how many pounds has he left? 

5. How many firkins will 3641b. of butter fill? Ans. 64. 

6. How many pounds in 17} quintals of codfish ? 

7. Bought 50 quintals of fish, at 6 cents a pound; what 


do they cost? Ans. $300. 
8. If half a barrel of flour is sold for $2.94, what is the 
price per pound? Ans, 3¢. 


9. How many more pounds are there in 20 barrels of 
salted beef than in 20 barrels of flour? 

10. If a barrel of pork brings $11.50, how much is that 
a pound ? Ans, 5c. 

11. How many ounces in 5 stone? : 

196. Recite the Miscellaneous Table. Is the pound here spoken of, the 


Avoirdupois or Troy pound? Is dried salted fish sold by the barrel or 
quintal? Is fish in pickle sold by the barrel or quintal? 


LONG MEASURE. » 115 


Lomg Measure. 


197. Long Measure is used in measuring length 
and distance. It begins with the inch. | 


1 inch. 
TABLE. 
12 inches (in.) make 1 foot, .. . ft. 
&~ feet, LOM er Cay ache TCL, 
5$ yards, Eerod ys. dacerd. 
40 rods, - 1 furlong, . fur. 
8 furlongs, Pamiles . mi! 


198. It is well to remember that (8x40) 820 rods or. 
(320 x 54x38) 5280 feet make a mile. 

199. The Hand, used as a measure of the height of horses, 
is 4inches. The Fathom, used as a measure of depths at sea, 
is 6 feet. 


MENTAL EXEROISES. 


. How many inches will a man 6 feet high measure? 
. If a horse is 15 hands high, how many feet is that? 
. How many feet in 10 yards? In 2 rods? 

. Reduce 4ft. 8in. to inches, 

How many inches in 2 fathoms? 

. How many yards in 6 rods? In 8 rods? 

. Reduce 108 inches to yards. 

. How many furlongs in 103 miles? 

. How many inches in a quarter of a yard? 


SOoanNTnar OD 


197. In what is Long Measure used? Drawa line aninchlong. Recite 
the Table.—198. How many rods make a mile? How many feet make a 
mile ?—199. What is the Hand used in measuring? How many inches make 
a hand? What is the Fathom used in measuring? How many feet make a 
fathom ? 


116 COMPOUND NUMBERS. 


EXAMPLES FOR THE SLATE. 
1. Reduce 108 yards to rods. 


5} yards make a rod; hence si must divide a yards by 
Gi: or 1. To divide by 4}, multiply 
a Migs by the ‘divisor inverted, ;%. Multi- 
11 )206 half-yd. plying by the numerator 2, and di- 
r18 Shf-yd. viding by the Sonor ee 11, we 
‘ ’ get 18 rods, and 8 remainder. 
Ans. 18rd. 4 yd. This remainder is not 8 yards, 
but 8 half-yards, since we reduced the original yards to half- 
yards when we multiplied by 2. Hence, to get the remainder 
in yards, divide it by 2. Answer, 18rd. 4yd. 
In reducing yards to rods, then, if there is @ remainder, 
divide it by 2, to bring it to yards. 


2. Reduce 464 yd. to rods. Ans, 84rd. 2yd. 
3. Reduce 1765 yd. to miles, &c. Ans. 1mi. 5 yd. 
4, Reduce 4855 in. to yards. Ans, 120 yd. 2 ft. 11 in. 
5. Reduce 248 mi. to inches. Ans, 15713280 in. 
6. How many miles, furlongs, &c., are there in 1051907 
inches ? Ans. 16 mi. 4fur. 32rd. 3yd. 1 ft. 11in. 
7. How many inches are there in 10mi. 1fur. 29rd. 
8 yd. 2ft. 10in.? Ans. 647404 in. 


8. Mt. Everest, a peak of the Himalayas, the highest moun- 
tain as yet surveyed, is 29002 feet above sea level. How 
many miles is this? Ans. 5mi. 2602 ft. 


200. In measuring drygoods, as cloth, muslin, 
lace, &c., the yard of long measure is used, divided 
into sige quarters, and eighths. 


1. What cost 19% yd. of lace, at 90c. a yard? 

2. How many pieces a quarter of a yard long can be cut 
from 12 yards of silk? 

3. If 2 dresses of 67,yd. and 103 yd. are cut from a piece ~ 
of calico containing 40 yd., how much is left ? 


SQUARE MEASURE. 117 


Square Measure, 


201. Square Measure is used in measuring surfaces, 
which have length and breadth; such as land, the 
sides of rooms, floors, &e. 

202. A Square is a figure that has four equal 
sides perpendicular one to an- 'Sagaen Teo. 
other—that is, leaning no more 
to one side than the other. 

A Square Inch is a square 
whose sides are each an inch long. 
A Square Foot is a square whose 
sides are each a foot long. 


TABLE. 


144 square inches (sq.in.), 1 square foot, sq. ft. 


9 square feet, 1 square yard, sq. yd. 
304 square yards, 1 square rod, sq.rd. 
40 square rods, Pr00d, a. tes able 

4 roods, APACT OS Fs co, Oks 

640 acres, 1 square mile, sq. mi. 


1. Reduce 40 A. 2R. tosquarerods. Ans, 6480sq.rd. 

2. Reduce 14245 sq.rd. to acres. Ans. 89. A. 5sq.rd. 

3. Reduce 3 sq. mi. to square rods. Ans. 307200sq. rd. 

4, How many acres in 59 lots, of 2 roods each ? 

5. What will it cost to plaster four walls, each containing 
270 sq. ft., at 20 cents a square yard? Ans. $24. 

6. How many farms of 82 acres will 1 sq. mi. make? 


201. In what is Square Measure used ?—202. What is a Square? What 
is a Square Inch? A Square Foot? Recite the Table. 


118 COMPOUND NUMBERS. 


Cubic Measure. 


203, Cubic Measure is used in measuring bodies, 
which have length, breadth, and depth or thickness ; 
as stone, timber, earth. 

204. A Cube is a body bounded by six equal 
squares. 

A Cubic Inch is a cube, one inch long, one inch 
broad, and one inch thick. Each of its six sides is 
a square inch. 

205. The engraving repre- 
sents a Cubic Yard. It is 
1 yard, or 3 feet, in length, 
_ breadth, and depth. It will 
be seen that each of its six 
sides is 1 square yard, or 9 
square feet. 

The top of this cube con- 
tains 9 square feet. Hence, 
if it were only 1 foot deep, it would contain 9 cubic 
feet. As itis 3 feet deep, it contains 3 times 9, or 
27, cubic feet. 


TABLE. 


1728 cubic inches (cu.in.), 1 cubic foot, cu. ft. 
27 cubic feet, 1 cubic yard, cu.yd. 
40 cu.ft. of round or 
50 cu. ft. of hewn timber, 1 tonordondyags 
16 cubic feet, 1 cord foot, . . ed. ft. 

8 cord feet, 1 cord; s sfonde 


Cubic Inch ?—205. What does the engraving represent? Describe it. How 
many cubic feet does it contain? Recite the Table. 


CUBIC MEASURE. © 119 


206. The ton in this Table is a measured ton; the Avoir- 
dupois ton is a ton of weight. Round timber is wood in its 
natural state. A ton of round timber consists of as much as, 
when hewn, will make 40 cubic feet. 

207. A cord of wood is a pile 8 feet long, 4 feet wide, 
and 4 feet high. Multiplying these dimensions together, we 
find 128 cubic feet in the cord. One foot in length of such a 
pile is called a cord foot. 

208. Cubic Measure is often used in estimating the 
amount of work in solid masonry, in digging cellars, making 
embankments, &c. 


EXAMPLES FOR THE SLATE. 


1. Reduce 9 cubic yards, 16 cubic feet, 862 cubic inches | 
to cubic inches. Ans. 448414 cu. in. 

2. Reduce 746496 cu. in. to cubic yards. Ans. 16. cu. yd. 

8. What will it cost to dig a cellar having a capacity of 
4860 cu. ft., at 22 cents a cubic yard? Ans. $39.60. 

Find the number of cubic yards ; multiply by the price. 

4, What will it cost to make an embankment containing 
108270 cu. ft. of earth, at 27c. a cubic yard? 

5. Find the price of 1536 cd. ft. of wood, at $5 a cord. 

6. How many cubic feet in 684 cords? 

7. Reduce 8 cu. yd. 469 cu. in. Ans. 373717 cu. in. 

8. How many feet of hewn timber in 472 tons? 


MISCELLANEOUS QueEstions.—What measure is used in finding 
the amount of surface in a floor? In estimating the amount of work 
in digging a well? In ascertaining the contents of a block of marble? 
In finding the distance between two places? In measuring one di- 
mension, such as length? In measuring what has two dimensions, 
length and breadth? In measuring what has three dimensions, 
length, breadth, and thickness? How does the ton of Cubic Measure 
differ from the Avoirdupois ton? What is round timber? How 
much round timber makes a ton? What is meant by a cord of 
wood? A cord foot? In what is Cubic Measure often used ? 


* 


120 COMFOUND NUMBERS. 


Liquid Measure, | 

209. Liquid or Wine Measure is used in measur- 

ing liquids generally; as, liquors (beer sometimes 
excepted), water, oil, milk, &c. 


Taste. | 

4 gills (gi.) make 1 pint,.. . pt. 

2 pints, 1 quart, . . qt. 
4 quarts, 1 gallon, . . gal. 
314 gallons, 1 barrel, . . bar. 
42 gallons, 1 ‘tierce, . tier, 
2 barrels (68 gal.), 1 hogshead, hhd. 

2 hogsheads, 1 pipe, “een 
2 pipes, 1 tun, Jeo 


210. The wine gallon contains 231 cubic inches. _ 

211. Liquids are put up in casks of different size, distin- 
guished as barrels, tierces, hogsheads, pipes, and tuns; but 
these casks do not uniformly contain the number of gallons 
assigned to them in the Table, but only about that quantity. 
The contents are found by gauging, or actual measurement. 
When the barrel is used in connection with the capacity of 
cisterns, vats, &c., 311 gallons are meant; in Massachusetts, 
32 gallons. 

MENTAL EXEROISES. 


1. How many gills in 2qt.? In 1 gallon? 

2. Reduce 3 qt. 1 pt. to gills. 

3. How often can you fill a quart measure from a four- 
gallon tub? From a six-gallon tub? 

4, What will a six-gallon can of milk cost, at 3c. a qt.? 

5. If a milkman sells 10 qt. out of a four-gallon can, how 
many quarts remain? 

6. How many gallons in 100 pints? 


WINE AND BEER MEASURE. 121 


Beer Measure. 
212. Beer Measure was formerly used in measur- 
ing beer and milk. It is still used by some for 
beer, though Wine Measure is fast taking its place. 


TABLE. — 
2 pints (pt.) make 1 quart, . . qt. 
4 quarts, 1 gallon,. . gal. 
36 gallons, 1 barrel, : . bar. 


1} barrels (54 gal.), 1 hogshead, hhd. 


213. The beer gallon contains 282 cubic inches. The 
gallon, quart, and pint of this measure, are therefore greater 
than those of Wine Measure. 


EXAMPLES FOR THE SLATE. 


1. Reduce 10000 gills to gallons. Ans. 312 gal. 2 qt. 

2. In 16 hogsheads of wine, of 63 gallons each, how many 
gills? Ans. 32256 gi. 

3. How many tuns will it take to hold 201600 gills of 
wine, allowing 252 gallons to the tun? Ans. 25 tuns. 

4. How many quarts in 42 tierces of oil, of 42 gal. each? 

5. Reduce 11 hhd. of beer to pints. 

6. In 100000 pints of beer, how many gallons? 

7, If a person buys a barrel of beer, containing 36 gallons, 
for $9, what does it cost him a quart? 

8. How many pints in 18 gal. 8qt. 1pt.? 

209. What is Liquid or Wine Measure used in measuring? Recite the 
Table.—210. low many cubic inches in a wine gallon ?—211. What is said 
of the different denominations, barrels, tierces, &c.? How are the contents 
of a cask ascertained? How many gallons go to the barrel in Massachusetts ? 
—212. For what was Beer Measure formerly used? For what is it still used ? 


Recite the Table.—213. How many cubic inches in a beer gallon? Which is 
greater, the beer quart or the wine quart ? 


122 - COMPOUND NUMBERS. 


Dry Measure. 


214. Dry Measure is used in measuring grain, 
seeds, vegetables, fruit, salt, coal, &e. 


TABLE. 


2 pints (pt.) make 1 quart, . . . qt. 


8 quarts, 1 peck, i pee 
4 pecks, 1 bushel, .. bu. 
36 bushels, 1 chaldron, . chal. 


215. The quart of Dry Measure is greater than that of 
Liquid Measure.—What is called the Small Measure contains 
2 qt. : 

216. Foreign coal is imported by the chaldron. American 
coal is bought and sold, in large quantities, by the ton; in 
small quantities, by the bushel. 


EXAMPLES FOR THE SLATE. 


. Reduce 56 bu. 2pk. 3 qt. to quarts. Ans. 1811 qt. 

. Reduce 8256 pt. to bushels. Ans, 129 bu. 

. How many bushels in 1212 chaldrons? Ans. 4880 bu. 

. Reduce 1597 qt. to bushels. Ans. 49bu. 3 pk. 5 qt. 

. How many small measures in a bushel? 

. How many chaldrons in 1848 bushels of coal? 

. Ifa bushel of apples is bought for 80c., and retailed at 

14¢c. a half-peck, what is the profit on them? Ans. 32¢. 
8. If 6 bushels of peaches are sold for $8.64, what do they 

bring a quart ? 


TI rnar ond & 


214. In whatis Dry Measure used? Recite the Table.—215. How does 
the quart of Dry Measure compare with that of Liquid Measure ?—216. What 
is imported by the chaldron? How is American coal bought and sold? 


MEASURE OF TIME. 1238 


Time Weasure. 


217. The natural divisions of time are the year 
and the day. The year is the period in which the 
Earth makes one revolution round the Sun; the 
day, that in which it makes one revolution on its 
axis. 

The year is divided into twelve calendar months, 
differing in length; the day, into hours, minutes, 
and seconds. 


TABLE. 
60 seconds (sec.) make 1 minute, .. min. 
60 minutes, TeHOnt uty as 
24 hours, As day fsa. “da, 
7 days, 1 week, . .. wk. 
12 calendar months or 
1 oorihen VTe 
365 days, Sa a 
366 days, 1 leap year. 
100 years, 1 century,. . cen. 


218. The twelve calendar months (mo.), and the 
number of days in each, are as follows:— 


DAYS. | DAYS. 
1st month, January, 81. | 7th month, July, 81, 
2d month, February, 28. | 8th month, August, 31. 
38d month, March, 31. | 9th month, September, 30. 
4th month, April, 30. | 10th month, October, 31. 
5th month, May, 31. | 11th month, November, 30. 
6th month, June, 80. | 12th month, December, 31. 


217. What are the natural divisions of time? Whatisthe year? What 
is the day? How is the year divided? The day? Recite the Table,—218. 
Name the twelve calendar months in order, and the number of days in each, 


124 COMPOUND NUMBERS. 


219. The days in these months, added together, . 
make 3865 days in the year. Every fourth year 
(except three in four centuries) is a Leap Year; 
then February has 29 days, and the year 366. 

220. The leap years are those that can be divided by 4 
without a remainder; as, 1864, 1868, 1872, &c. But, of the 
even hundreds, only those that can be divided by 400 are leap 
years. The year 1900 will not bea leap year, but 2000 will be. 

221. When we speak of a month, we mean a calendar month. 
The following lines will help the pupil to remember the num- 
ber of days in each :— 


‘‘ Thirty days hath September, 
April, June, and November ; 
All the rest have thirty-one, 
Except February alone; 
Which has but four and twenty-four, 
Till Leap Year gives it one day more.” 


EXAMPLES FOR THE SLATE. 


1. In 80 days how many seconds? Ans. 2592000 sec. 
2. Reduce 81920 min. to weeks. Ans. 8 wk. 21h. 20min. 
8. Find the number of seconds in 18wk. 3da. 19h. ° 


25 min. 39sec. Ans. 8191539 sec. 
4, How many seconds in 4 successive years, three common ~ 
years and one leap year ? Ans. 126230400 sec. 


5. Reduce 106847 sec. to hours. Ans. 29 h. 40min. 47sec. 

6. How often will a clock, ticking once a second, tick in 
24 hours? 

7. When 124 hours of aday have gone, how many minutes 
remain ? 


219. How often does Leap Year occur? How many days in a leap year? 
Which month receives the additional day ?—220. Which years are leap years? 
—221. Repeat the lines giving the number of days in the months. 


COMPOUND NUMBERS. 125 


Circular Tdeasure. 


222. Circular Measure is used chiefly in measur- 
ing angles and parts of circles, in determining lati- 
tude and longitude, and estimating the motions and 
positions of the heavenly bodies. 

223, Every circle may be divided into 360 equal 
parts, called Degrees. The actual length of the 
degree will of course depend on the size of the 
circle. The Sign is a division of the circle used 
only in Astronomy. 


TABLE. 
60 seconds (”), 1 minute,...’ 
60 minutes, 1 degree, i. 5° 
30 degrees, 1 sign, ilps 
12 signs (860°), 1 circlé,....C. 


Paper Measure. 


24 sheets make 1 quire. 

20 quires, 1 ream. 
2 reams, — 1 bundle. 
5 bundles, 1 bale. 


Collections of Wnits, 


12 units make 1 dozen, doz. 


12 dozen, 1 gross. 
12 gross, 1 great gross. 
20 units, 1 score. 


222. In what is Circular Measure chiefly used?—223. Into what may 
every circle be divided? On what will the actual length of the degree de- 
pend? Recite the Tables. 


« 


126 COMPOUND NUMBERS. 


EXAMPLES FOR THE SLATE. 


bed 
es 


Reduce 6 signs, 9 degrees, to degrees. 

In 1000 minutes how many degrees? 

. Reduce 38. 18° to seconds. Ans. 888800". 
. In 10000” how many degrees, &c.? Ans. 2° 46’ 40”. 
. Reduce 45° 45’ 85” to seconds. Ans. 164785”. 
In 1000 bottles of porter how many dozen? 

How many buttons in 182 dozen? 


ao ot RP © b 


. In 80064 tacks how many gross? 
. What will 480 bottles of ink cost, at $1 a dozen? 
10. Pens are put up in boxes containing a gross; how many 
pens in 5 dozen boxes? Ans. 8640 pens. 
11. If a box of pens is bought for 72 cents, what is the 
price of each pen? Ans. 5 mills. 
12. If a paper of tacks containing a grossis sold for 6 cents, 


0 0 7 


how many does that make for 1 cent? 

13. How many great gross in 1378944? 

14. How many sheets in one ream of paper? 

15. How many sheets in 153 quires? 

16. How many reams will 22480 sheets make? 

17. How many sheets in 27 reams, 3 quires? 

18. If half a ream of paper costs $2.40, what is the cost 
of each sheet? Ans. 1 ct. 

19. If a stationer buys paper at $2.50 a ream, and retails 
it at 20c. a quire, what does he make on a ream ? 

20. How much paper will it take to make 1000 books, 


containing 6 sheets each ? Ans, 121 reams. 
21. If 2 circulars are printed on a sheet, how many can be 
printed on 5 bundles of paper ? Ans. 9600 circulars. 


22. How many reams of paper will it take to print 9696 
such circulars? Ans. 10 reams, 2 quires. 


MISCELLANEOUS EXAMPLES, 127 


MISOELLANEOUS EXAMPLES IN COMPOUND NUMBERS. 


1. In 11959 grains, Apothecaries’ Weight, how many 
pounds, ounces, &c. ? Ans, 2tb. 73 19 er. 
2. The highest point of the globe ever attained by man is 
the top of Mt. Chimborazo, 19699 feet above sea level. What 


is this height in miles? Ans. 3mi. 8859 ft. 
3. Reduce 1 square mile, 29 acres, 8 square rods, 7 square 
yards, to square feet. Ans, 29148881 sq. ft. 


4, How many tumblerfuls, of half a pint each, will it take 
to fill a half-gallon pitcher ? 

5. Reduce 34 weeks to seconds. Ans, 2116800 sec. 

6. In 1000 cd. ft. of wood, how many cords? 

7. How many pages will there be in an edition of 2000 
books, each book made of 12 sheets, and each. sheet contain- 


ing 24 pages? Ans. 576000 pages. 
8. In 4lb. 50z. 1dr. 2sc. 10gr., how many grains are 
there? | Ans. 25550 gr. 


9. How many tons of hewn timber in 2500 cu. ft.? 
10. Reduce 85274 pt. to bushels. Ans. 1832 bu. 1 pk. 5 qt. 
11. How many lb. of silver will it take to make 4 dozen 


spoons weighing 18 pwt. each? Ans. 3lb. 7 0z. 4pwt. 
12. What will 15 firkins of butter come to, at 233 cents a 
pound? Ans. $197.40. 


13. How many doses of 6 drams each will 4 pounds of 
epsom salts make? 

14. A farmer wishes to put up 3671 bushels of potatoes in 
barrels holding 3 bu. 2 pk. each. How many barrels must he 


procure? Ans. 105 barrels. 


How many pecks are to be put up? How many pecks will each 
barrel hold ? 


15. At 2 cents a dozen, how much will a great gross of 
buttons cost? Ans. $2.88. 


= 


128 EXAMPLES IN COMPOUND NUMBERS. 


16. How many days are there in the Spring months, 
March, April, and May ? 

17. How many days in the Summer months, June, July, 
and August ? 

18. How many days in the Autumn or Fall months, Sep- 
tember, October, and November ? 


19. How many days in the Winter evig December, 
January, and February, when February falls in a leap year? 


20. Reduce 19 cu. ft. to cubic inches. Ans. 32832 cu. in. 

21. Reduce 5740 pwt. to pounds, Ans. 23]b. 11 0z. 

22. If from 3? of an ounce of gold a jeweller takes enough 
to make 6 rings weighing 25 pwt. each, how many penny- 
weights will he have left ? 3 

23. The depth of water at a certain spot is found to be 31 
fathoms, 3 ft. How many inches is this? Ans, 2268 in. 

24, How many brushes, at 2s. each, can be bought for 
£8? 

25. How long will 12 bushels of oats last a horse, if he is 
fed 8 quarts a day? 

26. At the rate of $4 a cont what is the value of a pile 
of wood, 8 feet long, 4 feet wide, and 4 feet high? 

27. If sound moves at the rate of 1120 feet in a second, 
how many miles off is a cannon that is heard 11 seconds after 
it is discharged ? Ans. 2mi. 1760 ft. 

28. If a family use 28]b. of flour in a week, how long 
will 2 barrels last them ? 

29. How many pounds sterling will 6 dozen combs cost, 


at 9d. apiece ? Ans. £2 14s. 
30. If a locomotive goes a mile in 2 minutes, how many 
hours will it take to go 150 miles? - Ans. 5h, 


31. Reduce 15 A. 3f. 20 sq. rods, 15 sq. yards, 2 sq. ft. 
to square inches. Ans. 99597888 sq. in. 


COMPOUND ADDITION. 129 


COMPOUND ADDITION. 


224. Compound numbers may be added, sub- 
tracted, multiplied, and divided. 


225. When compound numbers are added, the 
process is called Compound Addition. It combines 
addition and reduction ascending. 


226. A person spends in one store £6 5s. 7d.; in 
another, £7 1s. 2d. 1far.; in a third, £1 18s.; and 
in a fourth, £4 18s. 1d. 3far. How much does he 
lay out altogether ? 


We are here required to find the sum of several compound 
numbers. We must add things of the same kind; therefore 
write numbers of the same denomination in the same column. 
Mark the denominations over the top. 

Beginning at the right, add the first 
column. Its sum is 4 farthings, which, 


fo mo far. 

by dividing by 4, we reduce told. SetO 6 5 7 Q 
in the column of farthings, and carry 1 to 4 feta eae! igh 
the next column. 1218 4630 
The sum of the next column is 11. 4148 j{ 8 
11d. is not reducible to shillings, since 191711 0 


it takes 12d. to make 1s. Set down 11, 
therefore, under the column of pence. 

The sum of the shillings is 87s. = £1 17s. Set 17 under 
the column of shillings, and carry £1 to the next column. 

The sum of the next column is £19. As pounds can not 
be reduced to any higher denomination, we set 19 at once 
under the column added. Answer, £19 17s. 11d. Hence 
the following rule :— 


224, What operations may be performed on compound numbers ?—225, 
When compound numbers are added, what is the process called? What 
operations does Compound Addition combine ?—226. Explain the several 
steps in the given example, 


6* 


130 COMPOUND ADDITION. 


227. Rurz.—Zo add compound numbers, set them 
down so that the same denominations may stand in 
the same column. 

Beginning at the right, add the denominations 
separately. Set each sum under the column added, 
unless ut can be reduced to a higher denomination. 
If so, diwide by the number that tt takes to make 
one of that denomination; set the remainder under 
the column added, and carry the quotient. 

Proor.—Prove the addition, by adding in the 
opposite direction. 


EXAMPLES FOR THE SLATE. 


Add the following compound numbers. Always mark the 
denominations over the top. 


(1) (2) (8) 
£. 6 d, Ib. oz pwt. gr. is ae 3 5 ea 
Bec Se. Be Ti BED oh 2211". Fee 
8 1 11 2 17 22 10° Sate 
Qe. Oo 14 40 O° 2020 14 10" 2 ee 
18 0 112? 6 18 16 6.2950 
BE GEG 2° 108 16° 27 (” Dera 
384 14 8 RS Wise RP 386 6 6 «0 

(4) (5) 

T. <owt, qr.” Ib. oz, dr, mi, for, rdi~ ya. Teen 
OS OES See Epes =k 2.6 SY, eT as 
1 15 0 20 14 15 8-0 80) O72 
La henO nha b OSes nO 1 4° °O S56 ae 
Lo MS yy uae ee T° 130 eee 
FRE tek age 3 ino ER 2 °O 92" eel ae 
AGU PE SR Ee BNGE By 21° Sisk ae 


227. Recite the rule for Compound Addition, How may the addition be 
proved f © 


EXAMPLES IN COMPOUND ADDITION. 181 


(6) (7) (8) 
eq. yd. sq.ft. sq. in. cu. yd. cu. ft. cu.in. Cd. _ cd. ft. 
100 8 130 . 26 1000 3 7 

50 0 100 ag! 10 1541 10 4. 
10 5 0 20 80 12 1 
8 143 10 17 pid tas 8 6 

13 2 8 8 25 59 15 3 
175 7 93 26 18 963 50 5 

(9) (10) 

pi. hhd. gal, qt. pt. tun pi. hhd. gal. qt. pt. gi. 
1 bt 80.8.1 ey ncoks-. wikivctl ss bhi, Gaeg eee 
302420... Le 4 $016,.050.0554 60 19 Os ae 

25 oa 0 iB ale dog | EAS aaa Ws Ra S SAG 3 
00 Lied Mee De OE 25 ts 2. OL eG 
3 eS alae: eae | Gn -G. Ev nee! O14 eos 

(11) » (12) (13) 

bu. pk. qt. pt da, h. min. sec. ° . ” 
kia acti Pag 15 18 50 49 138 10 19 
m5 0 0 eye. Done 1 40 385 
5/2 38 1 4 23 47 2 2 48 39 
ot oe Rae | 2 10 15 30 40 
15 2 4 0 LOI Ree 4 10 45 45 


14. Find the sum of 2hhd. 50 gal. 3qt. 1lpt. (Beer 
Measure); 10hhd. 30 gal. 1qt.; 11hhd. 25 gal. 1pt.; 25 
hhd. Ae 1 qt.; and 6 hhd. 52 gal. 3 qt. 1 pt. 

Ans. 56bhd. 52 gal. 1 qt. 1 pt. 

15. Add together £7 18s. 8d.; £3 5s. 10d. 2far.; £6 18s. 
"d.; 2s. 5d. 8far.; £4 3d.; and £17 15s, 4d. 2 far. 

Ans. £39 15s. 9d. 3 far. 

16. Add together 1dr. 18 gr.; 2dr. 1sc. 15 gr.; 3dr. 2se. 
18 gr.; 4dr.; and 6dr. 1sc. 7 gr. Ans. 20z. 2dr. 138 gr. 

17. A stable-keeper uses 8 bu. 2 pk. of oats, one day; 7 bu. 
3 pk. 7qt., the next; Tbu. 6qt., the third; 6bu. 2 pk. 1pt., 


132 EXAMPLES IN COMPOUND ADDITION. 


the fourth; 8 bu., the fifth; 7bu. 1pk. 6qt. 1pt., the sixth; - 
and 6bu. 3pk. 5qt., the seventh. How much does he use 
during the week ? Ans. 52bu. 2pk. 1 qt. 
18. If one piece of cloth contains 40 yd.; another, 397 
yd.; a third, 383 yd.; a fourth, 393 yd.; and a fifth, 402 yd.; 
how many yards are there in all? Ans. 198} yd. 
19. What is the weight of four lots of iron, the first 
weighing 4cwt. 3 qr. 201b.; the second, 2T. 5cwt. 14]b.; 
the third, 1T. 2cwt. 2qr.;. and the fourth, 10T. 19 cwt. 
1 qr. 24 1b.? Ans. 14T. 12 ewt. 8 Ib. 
20. How much paper will a printer use for three jobs, if 
the first job requires 6 bales, 1 bundle; the second, 4 bun- 
dles, ream, 15 quires; and the third, 2 reams, 10 quires, 
12 sheets? Ans. 7 bales, 2 bundles, 15 quires, 12 sheets. 
21. A person owns five farms. The first contains 100A. 
1B. 30 sq. rd.; the second, 600 A. 2R. 10sq.rd.; the third, 
40 A. 1R. 12sq.rd.; the fourth, 250 A. 3R. 2sq.rd.; the 
fifth, 144 A. 20sq.rd. How much land does he own in all? 
Ans. 1186 A. 34 8q. rd. 
22. A manufacturer makes four lots of pens. The first 
consists of 20 great gross, 7 gross, 5 dozen, and 6; the second, 
of 9 gross, 10 dozen, and 5; the third, of 15 great gross, 11 
dozen; and the fourth, of 17 great gross, 3 gross. What is 
the whole amount made? 
Ans. 53 great gross, 9 gross, 2 dozen, and 11. 
23. Find the sum of 1 wk. 2 days 13h. 40 min. 30 SEC. 
2wk. 6 days 10h. 8 min. 8 sec.; 5 days 22 h. 55 min, 45 sec. ; 
4h. 1min. 15 sec.; and 1wk. 2 days 4h. 5 min. 
Ans. 6 wk. 8 days 6h. 50min. 383 sec. 
24. Add together 10rd. 4 yd. 2 ft. 8in.; 1rd. 3 yd. 5 in.; 
Srd. 2yd. 1ft. 6in.; 1rd. 4in.; and 2yd. 1ft. 9in. 
; Ans, 22rd. 2 yd. 8in. 


COMPOUND SUBTRACTION. 133 


COMPOUND SUBTRACTION. 


228. When one compound number is taken from 
another, the process is called Compound Subtraction. 
It combines subtraction and reduction descending. 


229. A person who had lewt. 8lb. 15 oz. of 
cheese, sold 3 qr. 191b. Toz. How much had he 
left ? 


We are here required to find the difference between two 
compound numbers. Write the subtrahend under the minuend, 
placing numbers of the same denomination in the same column, 
Mark the denominations over the top. 

Begin to subtract at the right. Toz.  cwt. qr Ib. oz 
from 150z. leave 80z.; set down 8 1 0 8 15 
beneath, in the same column. 19]b. Fs Resa bo Way § 
can not betaken from 8lb. Wetherefore 4ng SY ae: 
take one of the next higher denomina- 
tion, 1 qr., reduce it to pounds, andadditto 8lb. 25+8=33; 
191b. from 33 1b. leave 141b. Set down 14. 

To balance the quarter added to the minuend, we must 
add 1 qr. to the subtrahend. This we do, by carrying 1 to 
the next column. 

1 and 3 are 4. 4qr. can not be taken from Oqr. We 
therefore take 1 cwt., reduce it to quarters, and add it to 0 qr. 
44+0=4qr. .4qr. from 4qr. leave Oqr. Carry 1. 1from 
1, 0. Answer, 14]1b. 80z. Hence the rule. 


230. Rutr.—TZo subtract a compound number, 
set it under the minuend, placing numbers of the 
same denomination in the same column. 


228. When one compound number is taken from another, what is the 
process called? What does Compound Subtraction combine ?—229. Go 
through the example, explaining the steps.—230. Give the rule for Compound 
Subtraction, — 


134 COMPOUND SUBTRACTION. 


— Beginning at the right, subtract each denomina- 
tion separately, and place the remainder in the same 
column with the number subtracted. | 

Lf, in any denomination, the subtrahend is greater 
than the minuend, add to the latter as many as make 
one of the next higher denomination. Subtract, and 
carry 1 to the next denomination of the subtrahend. 

Proor.—Add the remainder and subtrahend. 
Lf the sum is equal to the minuend, the work is 


right. 
x EXAMPLES FOR THE SLATE. 
(1) (2) 
yr. mo. wk. da. T.. ewt.. qr: Ib. 62. dr, 
From 17 8 3 1 13.18. 1.200. 70eais 
Take 4: LZ. 6 10... 9. :3.. 23 sbaeane 
Ata. 33827 2D ee 3, 17 1.23 vais 
(3) (4) (5) 
tb 5.3 <@r" ee, mi, fur. rd. A. R. sq.rd, 
24 7 2 1 16 60 0 90 69 8 25 
16 340.4 Bi B17, 40 7 89 10 Q 88 
86d hd 19° O-uixd. 59 2 27 
(6) (7) (8) 
cu.yd. cu.ft. cu.in, ch. bu. pk. tun pi. hhd. gal. qt. 
144 12 123 30 10 1 10) "2 42. Spe 
89 23 869 10%.8::-8 1 0 0 60 8 


9. A grocer buys 15 cwt. 20 1b. of sugar, and sells 10 cwt. 


231b. How much remains unsold ? 


Ans, 4cwt. 3qr. 22 1b. 


10. From a piece of cloth containing 3872 yd., have been 


cut off 6} yd. for one dress and 10, yd. for another. 


many yards remain in the piece? 


How 
Ans. 21% yd. 


What is the proof in Compound Subtraction ? 


EXAMPLES IN COMPOUND SUBTRACTION. 135 


11. A person having £20 18s., spends £5 18s. 73d. How 
much has he left? Ans. £14 19s. 44d. 
12. A farmer raises 100 bu. 3 pk. 2 qt. of wheat from one 
field, and 87bu. 1pk. 1 qt. 1pt. from another. He sells 
58 bu. to one person, and 387 bu. 2 pk. 1 qt. to another. How 
much has he left? Ans. 97 bu. 2pk. 2.qt. 1 pt. 
18. From a pile of wood containing 100 cords, I sold 
10 Cd. 100 cu. ft. to one customer, and 18Cd. 59 cu. ft. to 
_ another. How many cords remained? Ans. 70Cd. 97 cu. ft. 

14. The subtrahend is 19 mi. 7 fur. 8rd. 2ft. 10 in.; the 
minuend is 24mi. 5fur. 18rd. 2yd. 1ft. 7in. What is the 
remainder ? Ans. 4mi. 6fur. 10rd. 1 yd. 1 ft. 9 in. 

15. A printer who has a bale of paper, uses 1 bundle, 
1 ream, 5 quires, and 6 sheets. How much remains on hand? 

16. A tailor uses 5 gross, 7 dozen, of buttons. How many 
has he on hand, if he had 1 great gross at first? 

17. Poughkeepsie is 75 miles from New York. A man 
who starts to walk there, goes 22mi. 8fur. 15rd. the first 
day, and 19 mi. 5fur. 80rd. the second. How much far- 
ther has he to go? Ans. 82 mi. 6 fur. 85rd. 

18. A jeweller, having a bar of silver weighing 2b. 6z., 
used 50z. 7pwt. 12 gr. for one job, and 1]b. 18pwt. 7 gr. 
for another. How much was left? Ans. 11 0z. 14 pwt. 5 gr. 

19. From a barrel of beer containing 54 gallons, a per- 
son drew 12 gal. 3 qt. one day, and 9gal. 2 qt. 1 pt. another. 
How much was left? 

20. From 39 sq. rd. 29 sq. yd. 128 sq. in., subtract 17 sq. rd. 
16 sq. yd. 5 sq. ft. 

21. A grocer has lewt. 18]b. of sugar in one barrel, 
38qr. 211b. in another, and lcwt. 2qr. 11]b. in a third. 
_ After selling lewt. 3qr. 15]b., how much will he have 
left ? Ans. lewt. 8 qr. 10]b. 


136 COMPOUND MULTIPLICATION. 


COMPOUND MULTIPLICATION. 


231. When a compound number is multiplied, 
the process is called Compound Multiplication. It 
combines multiplication and reduction ascending. 


232. Multiply 4bu. 3pk. Tqt. 1pt. by 27. 


27 being the product of 9 and 3, it is best to multiply by 
the factors in turn. Set 9, the first multiplier, under the ° 
lowest denomination of the multiplicand. 

Begin to multiply at the right. 9times >u. pk. qt. pt 
1 pint is 9 pt.; which, by dividing by 2, ee pi | 
we reduce to 4qt. 1 pt. Set 1 pt. in the 9 
column of pints, and cafry 4qt. to the 44 3 8 4 
next product. 3 

9 times. 7 qt. is 63 qt., and the 4qti. “4237 gee 
carried make 67 qt., equal to 8 pk. 3 qt. 1st -2 2 1 
Set 8 in the column of quarts, and carry 8 pk. to the next 
product. 

9 times 3pk. is 27 pk., and the 8 pk. carried make 35 pk., 
equal to 8 bu. 8pk. Set 3 in the column of pecks, and carry 
8 bu. to the next product. 

9 times 4 bu. is 86 bu., and the 8 bu. carried make 44 bu. 
Set down 44. 

Now multiply this product by 38, reducing and carrying in 
the same way. Answer, 134 bu. 2 pk. 2 qt. 1 pt. | 


233. Had we multiplied by 27 at once, we should haye 
proceeded in the same way. As, however, we could not then 
have multiplied and divided in the mind, we should have had 
to write out the figures elsewhere, and set the results only 
under the line.—Multiply by 12 or less at once, in one line. 
Sh 

231. When a compound number is multiplied, what is the process called ? 
What does Compound Multiplication combine ?—232.. Go through the ex. - 
ample, explaining the steps.—233. Had we multiplied by 27 at once, how 
should we have proceeded? How must we multiply by 12 or less? 


COMPOUND MULTIPLICATION. 137 


234, Rurz.—Se the multiplier under the lowest 
denomination of the multiplicand. 

Multiply each denomination in turn, and set the 
product under the number multiplied, unless it can 
be reduced to a higher denomination. If so, divide 
at by the number that a takes to make one of that 
denomination ; set the remainder under the num- 
ber multiplied, and carry the quotient to the next 
product. 

Proor.— When factors have been used in multi- 
plying, multiply by the factors in reverse order. 
Lf the results agree, the work ws right. 


EXAMPLES FOR THE SLATE. 


(1) (2) (3) 
Vd. ft:- mm. Sat Be d. oz. pwt. gr. 
Multiply 1 0 9 10 10 10 S72 2f oD 
By 4 3 spat f 
Ans. 4 3 0O SERA 4G 60 2 15 14 
(4) (5) 
ey wt, Ot, Ib.> oz. dr. hhd. gal. qt. pt. gi. 
OS Or Be 4n° 6 Ca ce: tee 
6 17 
Dee ee O18: Oo 14 17 49 3 12 


6. Multiply 5 cubic yards, 21 cubic feet, 648 cubic inches 


by 12. Ans. 69 cu. yd. 13 cu. ft. 804 cu. in. 
7. How much cloth will it take for 7 suits of clothes, if 
each suit requires 7 yd. 3 qr.? Ans. 54 yd. 1 qr. 


8. How much wood can a horse draw in 13 loads, if he 
draws 1Cd. 1 cd. ft. each load? 


234. Recite the rule for Compound Multiplication. What is the mode of 
proof, when factors have been used in multiplying ? 


138 COMPOUND MULTIPLICATION. 


9. How long will a man be in sawing 6 cords: of wood, if 
he takes 7 h. 30 min. 45 sec. to saw 1 cord, allowing 10 work- 
ing hours to each day? Ans. 4 days 5h. 4min. 80sec. 


Multiply in the usual way ; then reduce the hours to working days 
by dividing by 10. 


10. How much sugar is there in 21 hhd., each containing 
11 cwt. 3 qr. 15 lb.? Ans. 12T. 9cwt. 3 qr. 15 1b. 
11. Bought 15 yd. of broadcloth, at £1 8s. 6d. a yard, and 
22 yd. of silk, at 7s, 8d. 2far.a yd. What was the amount of 


the bill? Ans, £26 2s. 1d. 
Find the cost of each item; then add. 


12. The exact time in a year is 865 days 5h. 48 min. 
49,7, sec. What is the exact time in 50 years? 
(50=5 x10) Ans. 18262 days 2h. 41 min. 25sec. 
13. How much brandy in 84 pi., each containing 128 gal. 
2qt. 1 pt. 3 gi.? Ans, 10812 gal. 1 qt. 1 pt. 
14. If a man owning 5 farms, of 120A. 1R. 12 sq. rd. 
each, sells 450A. 3R. 25sq.rd., how much land has he 
left ? Ans. 150 A. 2R. 35 sq. rd. 
15. Bought 17 boxes of-raisins, at 12s, 4d. a box; 5 bar. 
of flour, at £1 10s. 6d. a barrel; and 16]b. of tea, at 5s. 8d. 
alb. Paid on account £19 10s.; how much remains un- 


paid ? Ans. £2 16s. 6d. 
16. What will be the yield of 82 acres of wheat, at the 
rate of 24bu. 2pk. 7 qt. per acre? Ans. 791 bu. 


17. If 2gal. 2qt. 1 pt. 1gi. leak out of a water pipe in 1 
hour, what will be the waste in 1 day? Ans. 68 gal. 8 qt. 
18. Suppose a person to walk, on an average, 3 mi. 2 fur. 
every morning, and 8 mi. 20rd. Iyd. every afternoon; how 
far will he walk.in two weeks? Ans. 88 mi. 8 fur. 2rd, 3 yd. 
19. If from 2 1b. of silver enough is taken to make a dozen 
spoons, weighing loz. 10pwt. 2gr. each, how much will be 
left ? . Ans. 5 oz. 19 pwt. 


COMPOUND DIVISION. 139 


COMPOUND DIVISION. 


~ 9835. When a compound number is divided, the 
process is called Compound Division, It combines 
division and reduction descending. 


236. Divide 148 gal. 3qt. 1 pt. 3 gi. by 23. 


Here we must use Long Division. Remember that a quo- 
tient is of the same denomination as the dividend from which 
it arises. Begin to divide at the left. 

Divide 148 gal. by 23: quo- Paar cre git 
tient, 6 gal.; remainder, 10 gal. 
; 23)148 3.1 8 (6 gal. 
Reduce the remainder to qt., 138 
and add in the 3 qt. in the divi- 


dend. 10x4=40 404+3=43. Shoe 
Divide 48 qt. by 28: quo- 23) 43 qt. (1 qt. 
tient, 1qt.; remainder, 20 qt. 23 
Reduce the remainder to pt., 20 qt. 
and add in the 1pt. in the | 
dividend. 20x2=40 40+1 23) 41 pt. (1 pt. 
—Af 23 
Divide 41 pt. by 23: quo- 18 pt. 
. tient, 1pt.; remainder, 18 pt. res 
Reduce the remainder to gi, 23) 75 gi. (Bay gi 
and add in the 8 gi. in the divi- 69 
dend. 18x4=72 7243=%5. 6 gi. 


am, 

Divide 75 gi. by 23: quo- Ang, 6gal. 1 qt. 1 pt. 8,5; gi. 
tient, 3 gi.; remainder, 6 gi. 
As thera i is no lower denomination to reduce hi remainder 
to, write it over the divisor in the form of a fraction, 5 

Collect the several quotients for the answer. 

If in any case there is no remainder, bring down the next 
denomination of the dividend, and proceed as above. 


235. When a compound number is divided, what is the process called ? 
What does Compound Division combine ?—236. Go through the example, ex- 
plaining the steps. 


td 
140 COMPOUND DIVISION. 


If the divisor is not contained in any dividend, set 0 in 
the quotient for that denomination, and reduce the dividend 
to the next. 

237. Rurze.—LBeginning at the left, diwide each 
denomination in turn. When there isaremainder, 
reduce ut to the next lower denomination, add in the 
number of that denomination in the gwen dividend, 
uf any, and continue the division. 

When there ts a remainder after the last division, 
place vt over the divisor, in the form of a fraction, 
and annex rt to the last quotient. The several quo- 
tients, each of the same denomination as its divi- 
dend, form the entire quotient. 

Proor.—Multiply the quotient by the divisor. 
Lf their product is equal to the dividend, the work 
as right. 3 

238, When the divisor is less than 12, use Short Division. 


EXAMPLES FOR THE SLATE. 


1) (2) 
mi. fur. rd. yd. ft. cwt, gr. Ib. *0oz, dr. 
pe ee a la ee 9)27 3B 17 =ia ae 
7 Bees Pareaae 9 Was @ Teas 8. 0.) 10 Sb ae 
. Divide 8 oz. 16 pwt. by 18. Ans. 13 pwt. 1242 gr. 
. Divide 22rd. 1yd.1ft.10in. by 11. Ans. 2rd. 53; in. 
. Divide £6 15s. 3d. by 10. Ans. 13s. 6d. 1} far. 


. Divide 86 bu. 1 pk. 1 pt. by 14. Ans. 6 bu. 5 qt. +% pt. 
. Divide 102 A.1R. 11 sq. rd. by 51. Ans. 2A. 1sq. rd. 
. Divide 4 gal. 2 qt. by 144. Ans. 1 gi. 
. Divide 40cu. yd. 10 cu. ft. by 18. 


Oo OT & Ct PR 


237. Recite the rule for Compound Division. What is the proof? 


EXAMPLES IN COMPOUND DIVISION. 141 


10. If 81 clocks cost £113 13s, 4d., how much is that 
apiece ? Ans. £8 18s, 4d. 
11. If 60z. 7dr. (Apothecaries’ Weight) of magnesia are 
put up in 60 equal parcels, how much will each weigh ? 
Ans, 2sc. 15 gr. 
12. A silversmith makes seven teapots, of equal weight, 
out of 9lb. loz. 14pwt. 5gr. of silver. What isthe weight 
of each ? Ans. 11b. 8 0z. 13 pwt. 11 er. 
13. If 47 casks, of the same size, hold 1686 gall. 1 pt. of 
beer, how much will each contain? Ans, 35 gal. 3 qt. 1 pt. 
14. If a traveller goes 600 miles in 1day 7h. 85 min. 20 
sec., what is his average time per mile? Ans. 3 min. 9;% sec. 
15. Divide 6 bales, 3bundles, lream, of paper into 8 
equal parts. Ans. 4bundles, 7 quires, 12 sheets. 
16. A farmer puts up 1000 bushels of apples in 350 barrels 
of uniform size. How many bushels, &c., does each barrel 
contain? — Te Ans. 2bu. 3pk. 32 qt. 
17. An estate worth £2570 is to be divided as follows: 
the widow is to have one third of the whole, and the rest is 
to be divided equally between seven children. What is the 
widow’s share, and what each child’s ? 


PAT Na. fa £856 13s. 4d. 
" (Child’s, £244 15s. 2d. 82 far. 


18. What is the weight of 13 crowns, each weighing 
18 pwt. 4,4 gr.; 14 shillings, each weighing 3 pwt. 15,3, gr.; 
and 9 sixpences, each weighing 1pwt. 19;> gr.? 

Ans. 1]b. 30z. 3pwt. 15 gr. 

19. A farmer having 450bu. 1pk. 1qt. of corn, after 
selling 425 bu. 3 pk. 6 qt., distributed the rest equally among 
5 poor families. How much did each receive? 

; Ans. 4bu. 3 pk. 8 qt. 13 pt. 
2°: Divide 182° 5’ 12” by 12. 


142 MISCELLANEOUS EXAMPLES. 


MISCELLANEOUS EXAMPLES. 


1. A person has $741.85 in one bank, $350 in another, 
and $1129.88 in a third; how much has he in bank al- 


together? 
-2. At 40 cents a yard, what will be the cost of 5 miles 
of telegraph cable? Ans. $3520. 


8. The silk-worm’s thread is about z,;5 of an inch thick; 
the spider’s web is about 3 as thick. How thick is it? 

4. Reduce 23 to its lowest terms. 

5. How many pens in 8,3; gross? 

6. Three boys, gathering nuts, agree to divide equally all 
they get. The first gathers 540, the second 960, the third 
720. How many does each receive? Ans. 740 nuts. 

7. Cut from a piece of cloth containing 36} yd., enough 
cloth to make 10 coats, each requiring 2} yd., and how much 
will be left? 

8. The product of two factors is 294. One of the factors 
is 83; what is the other ? Ans, 33. 

9. If 1 sovereign is worth $4.84, what are 27 sovereigns 
worth? 

10. A man who has } mile the start of another, gains on 
him 380 rods more. How many feet is he then ahead of the 
other ? 

11. If a boy who has been in the habit of sleeping 9h., 
rises an hour earlier every day, how many days will he save 


in 5 years, allowing for one leap year? Ans. 76da. 2h. 
12, From 2} subtract 13, and multiply the remainder 
by 34. - . Ans, 23. 


13. If a man has 8 small farms, of 8 fields each, and each 
field contains 2A. 3R. 22.sq.rd., how much land has he in 
all? Ans. 69A. 1R. 8 squrd. 


MISCELLANEOUS EXAMPLES, 143 


14, From a piece of cloth containing 121} yd., a tailor 
made 18 coats, which took one third of the whole piece. 
How many yards did each coat contain? Ans. 2} yd. 

15. How many pints in 263 gal. ? Ans. 2102 pt. 

16. A man worth $10000, at his decease, left $225.75 to 
the poor, and the rest to his five sons, in equal shares. How 
much did each get? Ans. $1954.85. 

"17. How many five-acre lots are there in a square mile? 

18. A trader, making four speculations, in the first gains 
$1520, in the second loses $460, in the third loses $720.50, 
and in the fourth gains $986.25. Does he gain or lose on the 
whole, and how much? Ans, Gains $1325.75. 

19. A man owning a mine, sells } of it to one person, 
2 of it to another, and % of it to a third. How much does 


he sell altogether ? Ans, 478, 
20. How many lots of half a rood each can be laid out 
in 63 acres? Ans. 52 lots. 


21. How many guineas are equivalent to £42? 

22. A man buys 7 head of cattle for $210. Two die, and 
he sells the rest for $34.50 apiece. How much does he lose? 

23. Add } of 90, 2 of 129, and 3 of 1260. Ans. 1001. 

24. If from 1 gal. of wine 3 qt. 1} pt. are sold, how much 
will remain? Ans. } pt. 

25. How many sheets of paper in three lots, consisting 
of 13 reams, 5 quires, and 1 bundle? Ans. 1800 sheets. 

26. Three persons, going on an excursion, agree to share 
the expense equally. They hire a carriage for $2.25, a boat 
for 75c. Their dinner costs them 50c. each, and their tea 
$1.20 for all four. How much has each to pay? Ans. $1.90. 

27. Which contains the most units, 44 dozen, 3 of a gross, 
or 5 Of a great gross? 

28. From ~ of 64 subtract 1 of 22. Ans. 4. 


144 MISCELLANEOUS EXAMPLES. 


29. How much is a man worth, over and above his debts, 
whose whole property is valued at £5100, and who owes one 
person £75 10s. and another £427 19s. 6d. 2 far. ? 

Ans. £4596 10s. 5d. 2 far. 

80. When a man who has 3} mi. to walk, has gone half- 
way, how many miles has he yet to go? 

31. How many dozen in } of a great gross? 8 

32. How many times is 8} contained in 91? Ans; Tt; 

83. A crown is worth 5s. How many crowns are equiva- 
lent to £25 10s. ? 

34. Two regiments, of 1125 men each, reach a river. 
They have 5 boats, capable of carrying 25 men each. How 
many trips must the boats make, to get them over? 

35. Reduce to a simple fraction 2 of 4 of 1 of 2% of $ 
of 3 of 12. Ans. 3. 

386. An officer, pursuing a thief, gains on him ¢ of a mile 
every hour. If the thief is 73 miles ahead, how many hours 


will it be before he is overtaken? Ans. 94h, 
37. How many more seconds does a leap year contain 
than an ordinary year? Ans. 86400 sec. 


88. A lad who has 288 marbles, loses § of them, sells }, 
and gives away ,3, of them. How many marbles has he 


left ? Ans. 48 marbles. 
39. How many cows, at $30 apiece, must a farmer give 
for 6 acres of land, at $15 an acre? Ans. 3 cows. 


Find the value of the land ; how many cows, at $30, will pay for it? 
40. A miller buys 22! cords of wood at $4 a cord, and 
pays for it in flour at $6 a barrel. How many barrels must 
he give? 
41. A weekly paper has been in existence 102 years. If 
each paper contains 8 pages, and each page 6 columns, how 
many columns has it contained during that time? 


» f{ suggestions from our best teachers, will be brought to bear, 


“UNIVERSITY OF ILLINOIS- URBANA ‘3 
| 51302E1866 


APPLETONS’ ARI AN FLEMENTARY ARITHMETIC. NEW YORK 


: ee Bouse: IIIINVININNN j 


UPON THE BASIS wx. sum VE Une Seda MRT oh 


| GEORGE R. PERKINS, LLD 


—————_——__9 o @-—_—_. + 


{ 


This New A Revie of Arithmeties will be as perfect in all res 
thought, and labor can make it. ‘All the extended experience 
and his peculiar faculty of imparting instruction to the -yo 


clear, comprehensive, philosop!ical, and practical system. ce 

The, different Numbers of this Series will be found x 
The advance from step to step is inductive and gradual; a 
‘pated, nothing required to be supplied by the teacher. The 
| simple, the rules brief, the analyses unencumbered with ver 
| amples drawn from the practical matters of life, the arra 
natural, the methods taught the shortest possible. Every 
to, to prevent, the mere mechanical doing of sums; the pu: 
stantly kept. on the alert, and his Arithmetic lesson i is thus 
able mental discipline. ' 


= 
of 


The Series will consist of the ne followinglll 


pk ‘A PRIMARY ARITHMETIC.—Beautifully, Hlusteated 

| ginner through the first four Rules and the simple Tables, ce 

examples with sums for the slate. Now ready. 16mo, 108 

i, AN ELEMENTARY ARYVIAMETIC—Reviews the sub a 

Piimary in a style adapted to somewhat iwaturer minds; ; 

Fractions, Yederal Money, Reduction, und the a Pi i 

. 12mo, 144 pages. 

Hit. A PRACTICAL ARITBMETIC—Full, cleat, conasanel 

| with diréet reference to the wants of Common Schools, anid | 

pare pupils thoroughly for tae business of life. Its me i 

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Iv. A HIGHER ARITHMETIC,—This work carries the s 

and contaifs’all that is required for a thorough mastery of th 

practice of Arithmetic. It is really what its name imports, In . 

¥. A MENTAL: ARITHMETIC.—For imparting read : mental: cal. 

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